Chapter 7: Powers, Roots, and Scientific Notation

VII

CHAPTER 7 Powers, Roots, and Scientific Notation

Square roots are used to determine the time it would take for a stone falling from the edge of this cliff to hit the land below.

This figure shows a rock cliff.

Suppose a stone falls from the edge of a cliff. The number of feet the stone has dropped after $t$ seconds can be found by multiplying 16 times the square of $t$ . But to calculate the number of seconds it would take the stone to hit the land below, we need to use a square root. In this chapter, we will introduce and apply the properties of exponents and square roots, and scientific notation.

7.1 Use Multiplication Properties of Exponents

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions with exponents
Simplify expressions using the Product Property for Exponents
Simplify expressions using the Power Property for Exponents
Simplify expressions using the Product to a Power Property
Simplify expressions by applying several properties
Multiply monomials

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, $2 to the 4$ means to multiply 2 by itself 4 times, so $2 to the 4$ means 2 · 2 · 2 · 2

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

This is read $a$ to the $m to the th$ power.

In the expression $a to the m$ , the exponent $m$ tells us how many times we use the base $a$ as a factor.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

EXAMPLE 1

Simplify: a) $4 to the 3$ b) $7 to the 1$ c) $(5 over 6) to the 2$ d) $(0.63) to the 2$ .

Solution

a)	$4 to the 3$
Multiply three factors of 4.	4 · 4 · 4
Simplify.	$64$
b)	$7 to the 1$
Multiply one factor of 7.	$7$
c)	$(5 over 6) to the 2$
Multiply two factors.	$(5 over 6)(5 over 6)$
Simplify.	$25 over 36$
d)	$(0.63) to the 2$
Multiply two factors.	$(0.63)(0.63)$
Simplify.	$0.3969$

TRY IT 1.1

Simplify: a) $6 to the 3$ b) $15 to the 1$ c) $(3 over 7) to the 2$ d) $(0.43) to the 2$ .

Show answer

a) 216 b) $15$ c) $9 over 49$ d) 0.1849

TRY IT 1.2

Simplify: a) $2 to the 5$ b) $21 to the 1$ c) $(2 over 5) to the 3$ d) $(0.218) to the 2$ .

Show answer

a) $32$ b) 21 c) $8 over 125$ d) $0.047524$

EXAMPLE 2

Simplify: a) $(-5) to the 4$ b) $-5 to the 4$ .

Solution

a)	$(-5) to the 4$
Multiply four factors of $-5$ .	$(-5)(-5)(-5)(-5)$
Simplify.	$625$
b)	$-5 to the 4$
Multiply four factors of 5.	-(5 · 5 · 5 · 5)
Simplify.	$-625$

TRY IT 2.1

Simplify: a) $(-3) to the 4$ b) $-3 to the 4$ .

Show answer

a) $81$ b) $-81$

TRY IT 2.2

Simplify: a) $(-13) to the 2$ b) $-13 to the 2$ .

Show answer

a) $169$ b) $-169$

Notice the similarities and differences in (Example 2) a) and (Example 2) b)! Why are the answers different? As we follow the order of operations in part a) the parentheses tell us to raise the $(-5)$ to the 4^th power. In part b) we raise just the 5 to the 4^th power and then take the opposite.

Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

What does this mean?
How many factors altogether?

x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.

So, we have

Notice that 5 is the sum of the exponents, 2 and 3.

x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.

We write:

$mathematical expression$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If $a$ is a real number, and $m$ and $n$ are counting numbers, then

$a to the m times a to the n=a to the m+n$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

$mathematical expression$

EXAMPLE 3

Simplify: $y to the 5 times y to the 6$ .

Solution


Use the product property, a^m \cdot aⁿ = a^m+n.
Simplify.

TRY IT 3.1

Simplify: $b to the 9 times b to the 8$ .

Show answer

$b to the 17$

TRY IT 3.2

Simplify: $x to the 12 times x to the 4$ .

Show answer

$x to the 16$

EXAMPLE 4

Simplify: a) $2 to the 5 times 2 to the 9$ b) $3 times 3 to the 4$ .

Solution

Use the product property, a^m · aⁿ = a^m+n.

Simplify.
Use the product property, a^m · aⁿ = a^m+n.

Simplify.

TRY IT 4.1

Simplify: a) $5 times 5 to the 5$ b) $4 to the 9 times 4 to the 9$ .

Show answer

a) $5 to the 6$ b) $4 to the 18$

TRY IT 4.2

Simplify: a) $7 to the 6 times 7 to the 8$ b) $10 times 10 to the 10$ .

Show answer

a) $7 to the 14$ b) $10 to the 11$

EXAMPLE 5

Simplify: a) $a to the 7 times a$ b) $x to the 27 times x to the 13$ .

Solution

Rewrite, a = a¹.

Use the product property, a^m · aⁿ = a^m+n.

Simplify.
Notice, the bases are the same, so add the exponents.

Simplify.

TRY IT 5.1

Simplify: a) $p to the 5 times p$ b) $y to the 14 times y to the 29$ .

Show answer

a) $p to the 6$ b) $y to the 43$

TRY IT 5.2

Simplify: a) $z times z to the 7$ b) $b to the 15 times b to the 34$ .

Show answer

a) $z to the 8$ b) $b to the 49$

We can extend the Product Property for Exponents to more than two factors.

EXAMPLE 6

Simplify: $d to the 4 times d to the 5 times d to the 2$ .

Solution


Add the exponents, since bases are the same.
Simplify.

TRY IT 6.1

Simplify: $x to the 6 times x to the 4 times x to the 8$ .

Show answer

$x to the 18$

TRY IT 6.2

Simplify: $b to the 5 times b to the 9 times b to the 5$ .

Show answer

$b to the 19$

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

What does this mean?
How many factors altogether?

x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.

So we have

Notice that 6 is the product of the exponents, 2 and 3.

x squared cubed is x to the power of 2 times 3, or x to the sixth power.

We write:

$mathematical expression$

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If $a$ is a real number, and $m$ and $n$ are whole numbers, then

$(a to the m) to the n=a to the m times n$

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

$mathematical expression$

EXAMPLE 7

Simplify: a) $(y to the 5) to the 9$ b) $(4 to the 4) to the 7$ .

Solution


Use the power property, (a^m)ⁿ = a^{m · n}.
Simplify.


Use the power property.
Simplify.

TRY IT 7.1

Simplify: a) $(b to the 7) to the 5$ b) $(5 to the 4) to the 3$ .

Show answer

a) $b to the 35$ b) $5 to the 12$

TRY IT 7.2

Simplify: a) $(z to the 6) to the 9$ b) $(3 to the 7) to the 7$ .

Show answer

a) $z to the 54$ b) $3 to the 49$

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

$(2x) to the 3$

What does this mean?

$2x times 2x times 2x$

We group the like factors together.

$2 times 2 times 2 times x times x times x$

How many factors of 2 and of $x$ ?

$2 to the 3 times x to the 3$

Notice that each factor was raised to the power and $(2x) to the 3$ is $2 to the 3 times x to the 3$ .

We write:	$(2x) to the 3$
	$2 to the 3 times x to the 3$

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents

If $a$ and $b$ are real numbers and $m$ is a whole number, then

$(ab) to the m=a to the mb to the m$

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

$mathematical expression$

EXAMPLE 8

Simplify: a) $(-9d) to the 2$ b) $(3mn) to the 3$ .

Solution

Use Power of a Product Property, (ab)^m = a^mb^m.

Simplify.
Use Power of a Product Property, (ab)^m = a^mb^m.

Simplify.

TRY IT 8.1

Simplify: a) $(-12y) to the 2$ b) $(2wx) to the 5$ .

Show answer

a) $144y to the 2$ b) $32w to the 5x to the 5$

TRY IT 8.2

Simplify: a) $(5wx) to the 3$ b) $(-3y) to the 3$ .

Show answer

a) $125w to the 3x to the 3$ b) $-27y to the 3$

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Properties of Exponents

If $a$ and $b$ are real numbers, and $m$ and $n$ are whole numbers, then

Product Property	$a to the m times a to the n =a to the m+n$
Power Property	$(a to the m) to the n = a to the m times n$
Product to a Power	$(ab) to the m = a to the mb to the m$

All exponent properties hold true for any real numbers $m$ and $n$ . Right now, we only use whole number exponents.

EXAMPLE 9

Simplify: a) $(y to the 3) to the 6(y to the 5) to the 4$ b) $(-6x to the 4y to the 5) to the 2$ .

Solution

a)	$(y to the 3) to the 6(y to the 5) to the 4$
Use the Power Property.	$y to the 18 times y to the 20$
Add the exponents.	$y to the 38$
b)	$(-6x to the 4y to the 5) to the 2$
Use the Product to a Power Property.	$(-6) to the 2(x to the 4) to the 2(y to the 5) to the 2$
Use the Power Property.	$(-6) to the 2$
Simplify.	$36x to the 8y to the 10$

TRY IT 9.1

Simplify: a) $(a to the 4) to the 5(a to the 7) to the 4$ b) $(-2c to the 4d to the 2) to the 3$ .

Show answer

a) $a to the 48$ b) $-8c to the 12d to the 6$

TRY IT 9.2

Simplify: a) $(-3x to the 6y to the 7) to the 4$ b) $(q to the 4) to the 5(q to the 3) to the 3$ .

Show answer

a) $81x to the 24y to the 28$ b) $q to the 29$

EXAMPLE 10

Simplify: a) $(5m) to the 2(3m to the 3)$ b) $(3x to the 2y) to the 4(2xy to the 2) to the 3$ .

Solution

a)	$(5m) to the 2(3m to the 3)$
Raise $5m$ to the second power.	$5 to the 2m to the 2 times 3m to the 3$
Simplify.	$25m to the 2 times 3m to the 3$
Use the Commutative Property.	$25 times 3 times m to the 2 times m to the 3$
Multiply the constants and add the exponents.	$75m to the 5$
b)	$(3x to the 2y) to the 4(2xy to the 2) to the 3$
Use the Product to a Power Property.	$(3 to the 4x to the 8y to the 4)(2 to the 3x to the 3y to the 6)$
Simplify.	$(81x to the 8y to the 4)(8x to the 3y to the 6)$
Use the Commutative Property.	$81 times 8 times x to the 8 times x to the 3 times y to the 4 times y to the 6$
Multiply the constants and add the exponents.	$648x to the 11y to the 10$

TRY IT 10.1

Simplify: a) $(5n) to the 2(3n to the 10)$ b) $(c to the 4d to the 2) to the 5(3cd to the 5) to the 4$ .

Show answer

a) $75n to the 12$ b) $81c to the 24d to the 30$

TRY IT 10.2

Simplify: a) $(a to the 3b to the 2) to the 6(4ab to the 3) to the 4$ b) $(2x) to the 3(5x to the 7)$ .

Show answer

a) $256a to the 22b to the 24$ b) $40x to the 10$

Multiply Monomials

A term in algebra is a constant or the product of a constant and one or more variables. When it is of the form $ax to the m$ , where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,-2x to the 2,4y to the 3$ , and $11z to the 7$ .

Monomials

A monomial is a term of the form $ax to the m$ , where $a$ is a constant and $m$ is a positive whole number.

Since a monomial is an algebraic expression,we can use the properties of exponents to multiply monomials.

EXAMPLE 11

Multiply: $(3x to the 2)(-4x to the 3)$ .

Solution

	$(3x to the 2)(-4x to the 3)$
Use the Commutative Property to rearrange the terms.	$3 times (-4) times x to the 2 times x to the 3$
Multiply.	$-12x to the 5$

TRY IT 11.1

Multiply: $(5y to the 7)(-7y to the 4)$ .

Show answer

$-35y to the 11$

TRY IT 11.2

Multiply: $(-6b to the 4)(-9b to the 5)$ .

Show answer

$54b to the 9$

EXAMPLE 12

Multiply: $(5 over 6x to the 3y)(12xy to the 2)$ .

Solution

	$(5 over 6x to the 3y)(12xy to the 2)$
Use the Commutative Property to rearrange the terms.	$5 over 6 times 12 times x to the 3 times x times y times y to the 2$
Multiply.	$10x to the 4y to the 3$

TRY IT 12.1

Multiply: $(2 over 5a to the 4b to the 3)(15ab to the 3)$ .

Show answer

$6a to the 5b to the 6$

TRY IT 12.2

Multiply: $(2 over 3r to the 5s)(12r to the 6s to the 7)$ .

Show answer

$8r to the 11s to the 8$

Additional Online Resources

Multiplication Properties of Exponents

Key Concepts

Exponential Notation
Properties of Exponents
- If $a,b$ are real numbers and $m,n$ are whole numbers, then
  $mathematical expression$

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

a) $3 to the 5$
b) $9 to the 1$
c) $(1 over 3) to the 2$
d) $(0.2) to the 4$

a) $10 to the 4$
b) $17 to the 1$
c) $(2 over 9) to the 2$
d) $(0.5) to the 3$

a) $2 to the 6$
b) $14 to the 1$
c) $(2 over 5) to the 3$
d) $(0.7) to the 2$

a) $8 to the 3$
b) $8 to the 1$
c) $(3 over 4) to the 3$
d) $(0.4) to the 3$

a) $(-6) to the 4$
b) $-6 to the 4$

a) $(-2) to the 6$
b) $-2 to the 6$

a) $-(1 over 4) to the 4$
b) $(-1 over 4) to the 4$

a) $-(2 over 3) to the 2$
b) $(-2 over 3) to the 2$

a) $-0.5 to the 2$
b) $(-0.5) to the 2$

10.

a) $-0.1 to the 4$
b) $(-0.1) to the 4$

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

11. $d to the 3 times d to the 6$	12. $x to the 4 times x to the 2$
13. $n to the 19 times n to the 12$	14. $q to the 27 times q to the 15$
15. a) $4 to the 5 times 4 to the 9$ b) $8 to the 9 times 8$	16. a) $3 to the 10 times 3 to the 6$ b) $5 times 5 to the 4$
17. a) $y times y to the 3$ b) $z to the 25 times z to the 8$	17. a) $y times y to the 3$ b) $z to the 25 times z to the 8$
19. $w times w to the 2 times w to the 3$	20. $y times y to the 3 times y to the 5$
21. $a to the 4 times a to the 3 times a to the 9$	22. $c to the 5 times c to the 11 times c to the 2$
23. $m to the x times m to the 3$	24. $n to the y times n to the 2$
25. $y to the a times y to the b$	26. $x to the p times x to the q$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression using the Power Property for Exponents.

27. a) $(m to the 4) to the 2$ b) $(10 to the 3) to the 6$	28. a) $(b to the 2) to the 7$ b) $(3 to the 8) to the 2$
29. a) $(y to the 3) to the x$ b) $(5 to the x) to the y$	30. a) $(x to the 2) to the y$ b) $(7 to the a) to the b$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

31. a) $(6a) to the 2$ b) $(3xy) to the 2$	32. a) $(5x) to the 2$ b) $(4ab) to the 2$
33. a) $(-4m) to the 3$ b) $(5ab) to the 3$	34. a) $(-7n) to the 3$ b) $(3xyz) to the 4$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

35.

a) $(y to the 2) to the 4 times (y to the 3) to the 2$
b) $(10a to the 2b) to the 3$

36.

a) $(w to the 4) to the 3 times (w to the 5) to the 2$
b) $(2xy to the 4) to the 5$

37.

a) $(-2r to the 3s to the 2) to the 4$
b) $(m to the 5) to the 3 times (m to the 9) to the 4$

38.

a) $(-10q to the 2p to the 4) to the 3$
b) $(n to the 3) to the 10cdot (n to the 5) to the 2$

39.

a) $(3x) to the 2(5x)$
b) $(5t to the 2) to the 3(3t) to the 2$

40.

a) $(2y) to the 3(6y)$
b) $(10k to the 4) to the 3(5k to the 6) to the 2$

41.

a) $(5a) to the 2(2a) to the 3$
b) $(1 over 2y to the 2) to the 3(2 over 3y) to the 2$

42.

a) $(4b) to the 2(3b) to the 3$
b) $(1 over 2j to the 2) to the 5(2 over 5j to the 3) to the 2$

43.

a) $(2 over 5x to the 2y) to the 3$
b) $(8 over 9xy to the 4) to the 2$

44.

a) $(2r to the 2) to the 3(4r) to the 2$
b) $(3x to the 3) to the 3(x to the 5) to the 4$

45.

a) $(m to the 2n) to the 2(2mn to the 5) to the 4$
b) $(3pq to the 4) to the 2(6p to the 6q) to the 2$

Multiply Monomials

In the following exercises, multiply the terms.

46. $(6y to the 7)(-3y to the 4)$	47. $(-10x to the 5)(-3x to the 3)$
48. $(-8u to the 6)(-9u)$	49. $(-6c to the 4)(-12c)$
50. $(1 over 5f to the 8)(20f to the 3)$	51. $(1 over 4d to the 5)(36d to the 2)$
52. $(4a to the 3b)(9a to the 2b to the 6)$	53. $(6m to the 4n to the 3)(7mn to the 5)$
54. $(4 over 7rs to the 2)(14rs to the 3)$	55. $(5 over 8x to the 3y)(24x to the 5y)$
56. $(2 over 3x to the 2y)(3 over 4xy to the 2)$	56. $(2 over 3x to the 2y)(3 over 4xy to the 2)$

Mixed Practice

In the following exercises, simplify each expression.

58. $(x to the 2) to the 4 times (x to the 3) to the 2$	59. $(y to the 4) to the 3 times (y to the 5) to the 2$
60. $(a to the 2) to the 6 times (a to the 3) to the 8$	61. $(b to the 7) to the 5 times (b to the 2) to the 6$
62. $(2m to the 6) to the 3$	63. $(3y to the 2) to the 4$
64. $(10x to the 2y) to the 3$	65. $(2mn to the 4) to the 5$
66. $(-2a to the 3b to the 2) to the 4$	67. $(-10u to the 2v to the 4) to the 3$
68. $(2 over 3x to the 2y) to the 3$	69. $(7 over 9pq to the 4) to the 2$
70. $(8a to the 3) to the 2(2a) to the 4$	71. $(5r to the 2) to the 3(3r) to the 2$
72. $(10p to the 4) to the 3(5p to the 6) to the 2$	73. $(4x to the 3) to the 3(2x to the 5) to the 4$
74. $(1 over 2x to the 2y to the 3) to the 4(4x to the 5y to the 3) to the 2$	75. $(1 over 3m to the 3n to the 2) to the 4(9m to the 8n to the 3) to the 2$
76. $(3m to the 2n) to the 2(2mn to the 5) to the 4$	77. $(2pq to the 4) to the 3(5p to the 6q) to the 2$

Everyday Math

78. Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is $10 to the 2$ , on the third round is $10 to the 3$ , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
1	10
2	$10 to the 2$
3	$10 to the 3$
…	…
6	?

79. Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $$35,000(1.03)$ , after 2 years was $$35,000(1.03) to the 2$ , after 3 years was $$35,000(1.03) to the 3$ , as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

Year	Salary
1	$$35,000(1.03)$
2	$$35,000(1.03) to the 2$
3	$$35,000(1.03) to the 3$
…	…
10	?

80. Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $$1,000(0.70)$ and the cost of the item during the second week would be $$1,000(0.70) to the 2$ . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week	Cost
1	$$1,000(0.70)$
2	$$1,000(0.70) to the 2$
3
…	…
5	?

81. Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for ?20,000, the value at the end of the first year would be $$20,000(0.90)$ and the value of the car after the end of the second year would be $$20,000(0.90) to the 2$ . Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Week	Cost
1	$$20,000(0.90)$
2	$$20,000(0.90) to the 2$
3
4	…
8	?

Writing Exercises

82. Use the Product Property for Exponents to explain why $x times x=x to the 2$ .	83. Explain why $-5 to the 3=(-5) to the 3$ but $-5 to the 4 not equal to (-5) to the 4$ .
84. Jorge thinks $(1 over 2) to the 2$ is 1. What is wrong with his reasoning?	85. Explain why $x to the 3 times x to the 5$ is $x to the 8$ , and not $x to the 15$

Answers

2. a) 10,000 b) 17 c) $4 over 81$ d) 0.125	4. a) 512 b) 8 c) $27 over 64$ d) 0.064
6. a) 64 b) $-64$	8. a) $-4 over 9$ b) $4 over 9$
10. a) $-0.0001$ b) $0.0001$	12. $x to the 6$
14. $q to the 42$	16. a) $3 to the 16$ b) $5 to the 5$
18. a) $w to the 6$ b) $u to the 94$	20. $y to the 9$
22. $c to the 18$	24. $n to the y+2$
26. $x to the p+q$	28. a) $b to the 14$ b) $3 to the 16$
30. a) $x to the 2y$ b) $7 to the ab$	32. a) $25x to the 2$ b) $16a to the 2b to the 2$
34. a) $-343n to the 3$ b) $81x to the 4y to the 4z to the 4$	36. a) $w to the 22$ b) $32x to the 5y to the 20$
38. a) $-1000q to the 6p to the 12$ b) $n to the 40$	40. a) $48y to the 4$ b) $25,000k to the 24$
42. a) $432b to the 5$ b) $1 over 200j to the 16$	44. a) $128r to the 8$ b) $1 over 200j to the 16$
46. $-18y to the 11$	48. $72u to the 7$
50. $4f to the 11$	52. $36a to the 5b to the 7$
54. $8r to the 2s to the 5$	56. $1 over 2x to the 3y to the 3$
58. $x to the 14$	60. $a to the 36$
62. $8m to the 18$	64. $1000x to the 6y to the 3$
66. $16a to the 12b to the 8$	68. $8 over 27x to the 6y to the 3$
70. $1024a to the 10$	72. $25000p to the 24$
74. $x to the 18y to the 18$	76. $144m to the 8n to the 22$
78. $1,000,000$	80. $168.07
82. Answers will vary.	84. Answers will vary.

Attributions

This chapter has been adapted from “Use Multiplication Properties of Exponents” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.2 Use Quotient Property of Exponents

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions using the Quotient Property for Exponents
Simplify expressions with zero exponents
Simplify expressions using the quotient to a Power Property
Simplify expressions by applying several properties

Simplify Expressions Using the Quotient Property for Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.

Summary of Exponent Properties for Multiplication

If $a$ and $b$ are real numbers, and $m$ and $n$ are whole numbers, then

Product Property	$a to the m times a to the n=a to the m+n$
Power Property	$(a to the m) to the n=a to the m times n$
Product to a Power	$(ab) to the m=a to the mb to the m$

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

Equivalent Fractions Property

If $a,b$ , and $c$ are whole numbers where $b not equal to 0,c not equal to 0$ ,

then $a over b=a times c over b times c$ and $a times c over b times c=a over b$

As before, we’ll try to discover a property by looking at some examples.

Consider

$x to the 5 over x to the 2$

and

$x to the 2 over x to the 3$

What do they mean?

$x times x times x times x times x over x times x$

$x times x over x times x times x$

Use the Equivalent Fractions Property.

$mathematical expression$

Simplify.

$x to the 3$

$1 over x$

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1

We write:

$mathematical expression$

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If $a$ is a real number, $a not equal to 0$ , and $m$ and $n$ are whole numbers, then

$a to the m over a to the n=a to the m-n,m$ > $n$ and $a to the m over a to the n=1 over a to the n-m,n$ > $m$

A couple of examples with numbers may help to verify this property.

$mathematical expression$

EXAMPLE 1

Simplify: a) $x to the 9 over x to the 7$ b) $3 to the 10 over 3 to the 2$ .

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Since 9 > 7, there are more factors of x in the numerator.

Use the Quotient Property, $a to the m over a to the n=a to the m-n$ .

Simplify.
Since 10 > 2, there are more factors of x in the numerator.

Use the Quotient Property, $a to the m over a to the n=a to the m-n$ .

Simplify.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

TRY IT 1.1

Simplify: a) $x to the 15 over x to the 10$ b) $6 to the 14 over 6 to the 5$ .

Show answer

a) $x to the 5$ b) $6 to the 9$

TRY IT 1.2

Simplify: a) $y to the 43 over y to the 37$ b) $10 to the 15 over 10 to the 7$ .

Show answer

a) $y to the 6$ b) $10 to the 8$

EXAMPLE 2

Simplify: a) $b to the 8 over b to the 12$ b) $7 to the 3 over 7 to the 5$ .

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Since 12 > 8, there are more factors of b in the denominator.

Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .

Simplify.
Since 5 > 3, there are more factors of 3 in the denominator.

Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .

Simplify.

Simplify.

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

TRY IT 2.1

Simplify: a) $x to the 18 over x to the 22$ b) $12 to the 15 over 12 to the 30$ .

Show answer

a) $1 over x to the 4$ b) $1 over 12 to the 15$

TRY IT 2.2

Simplify: a) $m to the 7 over m to the 15$ b) $9 to the 8 over 9 to the 19$ .

Show answer

a) $1 over m to the 8$ b) $1 over 9 to the 11$

Notice the difference in the two previous examples:

If we start with more factors in the numerator, we will end up with factors in the numerator.
If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

EXAMPLE 3

Simplify: a) $a to the 5 over a to the 9$ b) $x to the 11 over x to the 7$ .

Solution

Is the exponent of $a$ larger in the numerator or denominator? Since 9 > 5, there are more $a's$ in the denominator and so we will end up with factors in the denominator.

Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .

Simplify.
Notice there are more factors of $x$ in the numerator, since 11 > 7. So we will end up with factors in the numerator.

Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .

Simplify.

TRY IT 3.1

Simplify: a) $b to the 19 over b to the 11$ b) $z to the 5 over z to the 11$ .

Show answer

a) $b to the 8$ b) $1 over z to the 6$

TRY IT 3.2

Simplify: a) $p to the 9 over p to the 17$ b) $w to the 13 over w to the 9$ .

Show answer

a) $1 over p to the 8$ b) $w to the 4$

Simplify Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $a to the m over a to the m$ . From your earlier work with fractions, you know that:

$2 over 2=117 over 17=1-43 over -43=1$

In words, a number divided by itself is 1. So, $x over x=1$ , for any $x(x not equal to 0)$ , since any number divided by itself is 1

The Quotient Property for Exponents shows us how to simplify $a to the m over a to the n$ when $m$ > $n$ and when $n$ < $m$ by subtracting exponents. What if $m=n$ ?

Consider $8 over 8$ , which we know is 1

	$8 over 8=1$
Write $8$ as $2 to the 3$ .	$2 to the 3 over 2 to the 3=1$
Subtract exponents.	$2 to the 3-3=1$
Simplify.	$2 to the 0=1$

Now we will simplify $a to the m over a to the m$ in two ways to lead us to the definition of the zero exponent. In general, for $a not equal to 0$ :

$This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.$

We see $a to the m over a to the m$ simplifies to $a to the 0$ and to 1. So $a to the 0=1$ .

Zero Exponent

If $a$ is a non-zero number, then $a to the 0=1$ .

Any nonzero number raised to the zero power is 1

In this text, we assume any variable that we raise to the zero power is not zero.

EXAMPLE 4

Simplify: a) $9 to the 0$ b) $n to the 0$ .

Solution

The definition says any non-zero number raised to the zero power is 1

a) Use the definition of the zero exponent.	$mathematical expression$
b) Use the definition of the zero exponent.	$mathematical expression$

TRY IT 4.1

Simplify: a) $15 to the 0$ b) $m to the 0$ .

Show answer

a) 1 b) 1

TRY IT 4.2

Simplify: a) $k to the 0$ b) $29 to the 0$ .

Show answer

a) 1 b) 1

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let’s look at $(2x) to the 0$ . We can use the product to a power rule to rewrite this expression.

	$(2x) to the 0$
Use the product to a power rule.	$2 to the 0x to the 0$
Use the zero exponent property.	$1 times 1$
Simplify.	$1$

This tells us that any nonzero expression raised to the zero power is one.

EXAMPLE 5

Simplify: a) $(5b) to the 0$ b) $(-4a to the 2b) to the 0$ .

Solution

a)	$(5b) to the 0$
Use the definition of the zero exponent.	$1$
b)	$(-4a to the 2b) to the 0$
Use the definition of the zero exponent.	$1$

TRY IT 5.1

Simplify: a) $(11z) to the 0$ b) $(-11pq to the 3) to the 0$ .

Show answer

a) $1$ b) $1$

TRY IT 5.2

Simplify: a) $(-6d) to the 0$ b) $(-8m to the 2n to the 3) to the 0$ .

Show answer

a) $1$ b) $1$

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

	$(x over y) to the 3$
This means:	$x over y times x over y times x over y$
Multiply the fractions.	$x times x times x over y times y times y$
Write with exponents.	$x to the 3 over y to the 3$

Notice that the exponent applies to both the numerator and the denominator.

We write:	$(x over y) to the 3$
	$x to the 3 over y to the 3$

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If $a$ and $b$ are real numbers, $b not equal to 0$ , and $m$ is a counting number, then

$(a over b) to the m=a to the m over b to the m$

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

$mathematical expression$

EXAMPLE 6

Simplify: a) $(3 over 7) to the 2$ b) $(b over 3) to the 4$ c) $(k over j) to the 3$ .

Solution


Use the Quotient Property, $(a over b) to the m=a to the m over b to the m$ .
Simplify.


Use the Quotient Property, $(a over b) to the m=a to the m over b to the m$ .
Simplify.


Raise the numerator and denominator to the third power.

TRY IT 6.1

Simplify: a) $(5 over 8) to the 2$ b) $(p over 10) to the 4$ c) $(m over n) to the 7$ .

Show answer

a) $25 over 64$ b) $p to the 4 over 10,000$ c) $m to the 7 over n to the 7$

TRY IT 6.2

Simplify: a) $(1 over 3) to the 3$ b) $(-2 over q) to the 3$ c) $(w over x) to the 4$ .

Show answer

a) $1 over 27$ b) $-8 over q to the 3$ c) $w to the 4 over x to the 4$

Simplify Expressions by Applying Several Properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of Exponent Properties

If $a$ and $b$ are real numbers, and $m$ and $n$ are whole numbers, then

Product Property	$a to the m times a to the n=a to the m+n$
Power Property	$(a to the m) to the n=a to the m times n$
Product to a Power	$(ab) to the m=a to the mb to the m$
Quotient Property
Zero Exponent Definition	$a to the o=1,a not equal to 0$
Quotient to a Power Property	$(a over b) to the m=a to the m over b to the m,b not equal to 0$

EXAMPLE 7

Simplify: $(y to the 4) to the 2 over y to the 6$ .

Solution

	$(y to the 4) to the 2 over y to the 6$
Multiply the exponents in the numerator.	$y to the 8 over y to the 6$
Subtract the exponents.	$y to the 2$

TRY IT 7.1

Simplify: $(m to the 5) to the 4 over m to the 7$ .

Show answer

$m to the 13$

TRY IT 7.2

Simplify: $(k to the 2) to the 6 over k to the 7$ .

Show answer

$k to the 5$

EXAMPLE 8

Simplify: $b to the 12 over (b to the 2) to the 6$ .

Solution

	$b to the 12 over (b to the 2) to the 6$
Multiply the exponents in the numerator.	$b to the 12 over b to the 12$
Subtract the exponents.	$b to the 0$
Simplify.	$1$

TRY IT 8.1

Simplify: $n to the 12 over (n to the 3) to the 4$ .

Show answer

TRY IT 8.2

Simplify: $x to the 15 over (x to the 3) to the 5$ .

Show answer

EXAMPLE 9

Simplify: $(y to the 9 over y to the 4) to the 2$ .

Solution

	$(y to the 9 over y to the 4) to the 2$
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents.	$(y to the 5) to the 2$
Multiply the exponents.	$y to the 10$

TRY IT 9.1

Simplify: $(r to the 5 over r to the 3) to the 4$ .

Show answer

$r to the 8$

TRY IT 9.2

Simplify: $(v to the 6 over v to the 4) to the 3$ .

Show answer

$v to the 6$

EXAMPLE 10

Simplify: $(j to the 2 over k to the 3) to the 4$ .

Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

	$(j to the 2 over k to the 3) to the 4$
Raise the numberator and denominator to the third power using the Quotient to a Power Property, $(a over b) to the m=a to the m over b to the m$ .
Use the Power Property and simplify.

TRY IT 10.1

Simplify: $(a to the 3 over b to the 2) to the 4$ .

Show answer

$a to the 12 over b to the 8$

TRY IT 10.2

Simplify: $(q to the 7 over r to the 5) to the 3$ .

Show answer

$q to the 21 over r to the 15$

EXAMPLE 11

Simplify: $(2m to the 2 over 5n) to the 4$ .

Solution

	$(2m to the 2 over 5n) to the 4$
Raise the numberator and denominator to the fourth power, using the Quotient to a Power Property, $(a over b) to the m=a to the m over b to the m$ .	$(2m to the 2) to the 4 over (5n) to the 4$
Raise each factor to the fourth power.	$(2m to the 2) to the 4 over (5n) to the 4$
Use the Power Property and simplify.	$16m to the 8 over 625n to the 4$

TRY IT 11.1

Simplify: $(7x to the 3 over 9y) to the 2$ .

Show answer

$49x to the 6 over 81y to the 2$

TRY IT 11.2

Simplify: $(3x to the 4 over 7y) to the 2$ .

Show answer

$9x to the 8 over 49y to the 2$

EXAMPLE 12

Simplify: $(x to the 3) to the 4(x to the 2) to the 5 over (x to the 6) to the 5$ .

Solution

	$(x to the 3) to the 4(x to the 2) to the 5 over (x to the 6) to the 5$
Use the Power Property, $(a to the m) to the n=a to the m times n$ .	$(x to the 12)(x to the 10) over (x to the 30)$
Add the exponents in the numerator.	$x to the 22 over x to the 30$
Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .	$1 over x to the 8$

TRY IT 12.1

Simplify: $(a to the 2) to the 3(a to the 2) to the 4 over (a to the 4) to the 5$ .

Show answer

$1 over a to the 6$

TRY IT 12.2

Simplify: $(p to the 3) to the 4(p to the 5) to the 3 over (p to the 7) to the 6$ .

Show answer

$1 over p to the 15$

EXAMPLE 13

Simplify: $(10p to the 3) to the 2 over (5p) to the 3(2p to the 5) to the 4$ .

Solution

	$(10p to the 3) to the 2 over (5p) to the 3(2p to the 5) to the 4$
Use the Product to a Power Property, $(ab) to the m=a to the mb to the m$ .	$(10) to the 2(p to the 3) to the 2 over (5) to the 3(p) to the 3(2) to the 4(p to the 5) to the 4$
Use the Power Property, $(a to the m) to the n=a to the m times n$ .	$100p to the 6 over 125p to the 3 times 16p to the 20$
Add the exponents in the denominator.	$100p to the 6 over 125 times 16p to the 23$
Use the Quotient Property, $a to the m over a to the n=1 over a to the n-m$ .	$100 over 125 times 16p to the 17$
Simplify.	$1 over 20p to the 17$

TRY IT 13.1

Simplify: $(3r to the 3) to the 2(r to the 3) to the 7 over (r to the 3) to the 3$ .

Show answer

$9r to the 18$

TRY IT 13.2

Simplify: $(2x to the 4) to the 5 over (4x to the 3) to the 2(x to the 3) to the 5$ .

Show answer

$2 over x$

Divide Monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

EXAMPLE 14

Find the quotient: $56x to the 7 divided by 8x to the 3$ .

Solution

	$56x to the 7 divided by 8x to the 3$
Rewrite as a fraction.	$56x to the 7 over 8x to the 3$
Use fraction multiplication.	$56 over 8 times x to the 7 over x to the 3$
Simplify and use the Quotient Property.	$7x to the 4$

TRY IT 14.1

Find the quotient: $42y to the 9 divided by 6y to the 3$ .

Show answer

$7y to the 6$

TRY IT 14.2

Find the quotient: $48z to the 8 divided by 8z to the 2$ .

Show answer

$6z to the 6$

EXAMPLE 15

Find the quotient: $45a to the 2b to the 3 over -5ab to the 5$ .

Solution

	$45a to the 2b to the 3 over -5ab to the 5$
Use fraction multiplication.	$45 over -5 times a to the 2 over a times b to the 3 over b to the 5$
Simplify and use the Quotient Property.	$-9 times a times 1 over b to the 2$
Multiply.	$-9a over b to the 2$

TRY IT 15.1

Find the quotient: $-72a to the 7b to the 3 over 8a to the 12b to the 4$ .

Show answer

$-9 over a to the 5b$

TRY IT 15.2

Find the quotient: $-63c to the 8d to the 3 over 7c to the 12d to the 2$ .

Show answer

$-9d over c to the 4$

EXAMPLE 16

Find the quotient: $24a to the 5b to the 3 over 48ab to the 4$ .

Solution

	$24a to the 5b to the 3 over 48ab to the 4$
Use fraction multiplication.	$24 over 48 times a to the 5 over a times b to the 3 over b to the 4$
Simplify and use the Quotient Property.	$1 over 2 times a to the 4 times 1 over b$
Multiply.	$a to the 4 over 2b$

TRY IT 16.1

Find the quotient: $16a to the 7b to the 6 over 24ab to the 8$ .

Show answer

$2a to the 6 over 3b to the 2$

TRY IT 16.2

Find the quotient: $27p to the 4q to the 7 over -45p to the 12q$ .

Show answer

$-3q to the 6 over 5p to the 8$

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

EXAMPLE 17

Find the quotient: $14x to the 7y to the 12 over 21x to the 11y to the 6$ .

Solution

Be very careful to simplify $14 over 21$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

	$14x to the 7y to the 12 over 21x to the 11y to the 6$
Simplify and use the Quotient Property.	$2y to the 6 over 3x to the 4$

TRY IT 17.1

Find the quotient: $28x to the 5y to the 14 over 49x to the 9y to the 12$ .

Show answer

$4y to the 2 over 7x to the 4$

TRY IT 17.2

Find the quotient: $30m to the 5n to the 11 over 48m to the 10n to the 14$ .

Show answer

$5 over 8m to the 5n to the 3$

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

EXAMPLE 18

Find the quotient: $(6x to the 2y to the 3)(5x to the 3y to the 2) over (3x to the 4y to the 5)$ .

Solution

	$(6x to the 2y to the 3)(5x to the 3y to the 2) over (3x to the 4y to the 5)$
Simplify the numerator.	$30x to the 5y to the 5 over 3x to the 4y to the 5$
Simplify.	$10x$

TRY IT 18.1

Find the quotient: $(6a to the 4b to the 5)(4a to the 2b to the 5) over 12a to the 5b to the 8$ .

Show answer

$2ab to the 2$

TRY IT 18.2

Find the quotient: $(-12x to the 6y to the 9)(-4x to the 5y to the 8) over -12x to the 10y to the 12$ .

Show answer

$-4xy to the 5$

Additional Online Resources

Key Concepts

Quotient Property for Exponents:
- If $a$ is a real number, $a not equal to 0$ , and $m,n$ are whole numbers, then:
  $a to the m over a to the n=a to the m-n,m$ > $n and a to the m over a to the n=1 over a to the m-n,n$ > $m$
Zero Exponent
- If $a$ is a non-zero number, then $a to the 0=1$ .
Quotient to a Power Property for Exponents:
- If $a$ and $b$ are real numbers, $b not equal to 0$ , and $m$ is a counting number, then:
  $(a over b) to the m=a to the m over b to the m$
- To raise a fraction to a power, raise the numerator and denominator to that power.
Summary of Exponent Properties
- If $a,b$ are real numbers and $m,n$ are whole numbers, then

Summary of Product, Power, Product to a Power, Quotient, Zero Exponent Definition, and Quotient to a Power Properties.

Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

1. a) $x to the 18 over x to the 3$ b) $5 to the 12 over 5 to the 3$	2. a) $y to the 20 over y to the 10$ b) $7 to the 16 over 7 to the 2$
3. a) $p to the 21 over p to the 7$ b) $4 to the 16 over 4 to the 4$	4. a) $u to the 24 over u to the 3$ b) $9 to the 15 over 9 to the 5$
5. a) $q to the 18 over q to the 36$ b) $10 to the 2 over 10 to the 3$	6. a) $t to the 10 over t to the 40$ b) $8 to the 3 over 8 to the 5$
7. a) $b over b to the 9$ b) $4 over 4 to the 6$	8. a) $x over x to the 7$ b) $10 over 10 to the 3$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

a) $20 to the 0$
b) $b to the 0$

10.

a) $13 to the 0$
b) $k to the 0$

11.

a) $-27 to the 0$
b) $-(27 to the 0)$

12.

a) $-15 to the 0$
b) $-(15 to the 0)$

13.

a) $(25x) to the 0$
b) $25x to the 0$

14.

a) $(6y) to the 0$
b) $6y to the 0$

15.

a) $(12x) to the 0$
b) $(-56p to the 4q to the 3) to the 0$

16.

a) $7y to the 0$ $(17y) to the 0$
b) $(-93c to the 7d to the 15) to the 0$

17.

a) $12n to the 0-18m to the 0$
b) $(12n) to the 0-(18m) to the 0$

18.

a) $15r to the 0-22s to the 0$
b) $(15r) to the 0-(22s) to the 0$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

19.

a) $(3 over 4) to the 3$ b) $(p over 2) to the 5$ c) $(x over y) to the 6$

20.

a) $(2 over 5) to the 2$ b) $(x over 3) to the 4$ c) $(a over b) to the 5$

21.

a) $(a over 3b) to the 4$ b) $(5 over 4m) to the 2$

22.

a) $(x over 2y) to the 3$ b) $(10 over 3q) to the 4$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

23. $(a to the 2) to the 3 over a to the 4$	24. $(p to the 3) to the 4 over p to the 5$
25. $(y to the 3) to the 4 over y to the 10$	26. $(x to the 4) to the 5 over x to the 15$
27. $u to the 6 over (u to the 3) to the 2$	28. $v to the 20 over (v to the 4) to the 5$
29. $m to the 12 over (m to the 8) to the 3$	30. $n to the 8 over (n to the 6) to the 4$
31. $(p to the 9 over p to the 3) to the 5$	32. $(q to the 8 over q to the 2) to the 3$
33. $(r to the 2 over r to the 6) to the 3$	34. $(m to the 4 over m to the 7) to the 4$
35. $(p over r to the 11) to the 2$	36. $(a over b to the 6) to the 3$
37. $(w to the 5 over x to the 3) to the 8$	38. $(y to the 4 over z to the 10) to the 5$
39. $(2j to the 3 over 3k) to the 4$	40. $(3m to the 5 over 5n) to the 3$
41. $(3c to the 2 over 4d to the 6) to the 3$	42. $(5u to the 7 over 2v to the 3) to the 4$
43. $(k to the 2k to the 8 over k to the 3) to the 2$	44. $(j to the 2j to the 5 over j to the 4) to the 3$
45. $(t to the 2) to the 5(t to the 4) to the 2 over (t to the 3) to the 7$	46. $(q to the 3) to the 6(q to the 2) to the 3 over (q to the 4) to the 8$
47. $(-2p to the 2) to the 4(3p to the 4) to the 2 over (-6p to the 3) to the 2$	48. $(-2k to the 3) to the 2(6k to the 2) to the 4 over (9k to the 4) to the 2$
49. $(-4m to the 3) to the 2(5m to the 4) to the 3 over (-10m to the 6) to the 3$	50. $(-10n to the 2) to the 3(4n to the 5) to the 2 over (2n to the 8) to the 2$

Divide Monomials

In the following exercises, divide the monomials.

51. $56b to the 8 divided by 7b to the 2$	52. $63v to the 10 divided by 9v to the 2$
53. $-88y to the 15 divided by 8y to the 3$	54. $-72u to the 12 divided by 12u to the 4$
55. $45a to the 6b to the 8 over -15a to the 10b to the 2$	56. $54x to the 9y to the 3 over -18x to the 6y to the 15$
57. $15r to the 4s to the 9 over 18r to the 9s to the 2$	58. $20m to the 8n to the 4 over 30m to the 5n to the 9$
59. $18a to the 4b to the 8 over -27a to the 9b to the 5$	60. $45x to the 5y to the 9 over -60x to the 8y to the 6$
61. $64q to the 11r to the 9s to the 3 over 48q to the 6r to the 8s to the 5$	62. $65a to the 10b to the 8c to the 5 over 42a to the 7b to the 6c to the 8$
63. $(10m to the 5n to the 4)(5m to the 3n to the 6) over 25m to the 7n to the 5$	64. $(-18p to the 4q to the 7)(-6p to the 3q to the 8) over -36p to the 12q to the 10$
65. $(6a to the 4b to the 3)(4ab to the 5) over (12a to the 2b)(a to the 3b)$	66. $(4u to the 2v to the 5)(15u to the 3v) over (12u to the 3v)(u to the 4v)$

Mixed Practice

67. a) $24a to the 5+2a to the 5$ b) $24a to the 5-2a to the 5$ c) $mathematical expression$ d) $24a to the 5 divided by 2a to the 5$	69. a) $p to the 4 times p to the 6$ b) $(p to the 4) to the 6$
70. a) $q to the 5 times q to the 3$ b) $(q to the 5) to the 3$	71. a) $y to the 3 over y$ b) $y over y to the 3$
72. a) $z to the 6 over z to the 5$ b) $z to the 5 over z to the 6$	73. $(8x to the 5)(9x) divided by 6x to the 3$
74. $(4y)(12y to the 7) divided by 8y to the 2$	75. $27a to the 7 over 3a to the 3+54a to the 9 over 9a to the 5$
76. $32c to the 11 over 4c to the 5+42c to the 9 over 6c to the 3$	77. $32y to the 5 over 8y to the 2-60y to the 10 over 5y to the 7$
78. $48x to the 6 over 6x to the 4-35x to the 9 over 7x to the 7$	79. $63r to the 6s to the 3 over 9r to the 4s to the 2-72r to the 2s to the 2 over 6s$
80. $56y to the 4z to the 5 over 7y to the 3z to the 3-45y to the 2z to the 2 over 5y$

Everyday Math

81. Memory One megabyte is approximately $10 to the 6$ bytes. One gigabyte is approximately $10 to the 9$ bytes. How many megabytes are in one gigabyte?

82. Memory One gigabyte is approximately $10 to the 9$ bytes. One terabyte is approximately $10 to the 12$ bytes. How many gigabytes are in one terabyte?

Writing Exercises

83. Jennifer thinks the quotient $a to the 24 over a to the 6$ simplifies to $a to the 4$ . What is wrong with her reasoning?	84. Maurice simplifies the quotient $d to the 7 over d$ by writing $mathematical expression$ . What is wrong with his reasoning?
85. When Drake simplified $-3 to the 0$ and $(-3) to the 0$ he got the same answer. Explain how using the Order of Operations correctly gives different answers.	86. Robert thinks $x to the 0$ simplifies to 0. What would you say to convince Robert he is wrong?

Answers

2. a) $y to the 10$ b) $7 to the 14$	4. a) $u to the 21$ b) $9 to the 10$
6. a) $1 over t to the 30$ b) $1 over 64$	8. a) $1 over x to the 6$ b) $1 over 100$
10. a) 1 b) 1	12. a) $-1$ b) $-1$
14. a) 1 b) 6	16. a) 7 b) 1
18. a) $-7$ b) 0	20. a) $4 over 25$ b) $x to the 4 over 81$ c) $a to the 5 over b to the 5$
22. a) $x to the 3 over 8y to the 3$ b) $10,000 over 81q to the 4$	24. $p to the 7$
26. $x to the 5$	28. 1
30. $1 over n to the 12$	32. $q to the 18$
34. $1 over m to the 12$	36. $a to the 3 over b to the 18$
38. $y to the 20 over z to the 50$	40. $27m to the 15 over 125n to the 3$
42. $625u to the 28 over 16v to the ^12$	44. $j to the 9$
46. $1 over q to the 8$	48. $64k to the 6$
50. $-4,000$	52. $7v to the 8$
54. $-6u to the 8$	56. $-3x to the 3 over y to the 12$
58. $-2m to the 3 over 3n to the 5$	60. $-3y to the 3 over 4x to the 3$
62. $65a to the 3b to the 2 over 42c to the 3$	64. $-3q to the 5 over p to the 5$
66. $5v to the 4 over u to the 2$	68. a) $18n to the 10$ b) $12n to the 10$ c) $45n to the 20$ d) 5
70. a) $q to the 8$ b) $q to the 15$	72. a) $z$ b) $1 over z$
74. $6y to the 6$	76. $15c to the 6$
78. $3x to the 2$	80. $-yz to the 2$
82. $10 to the 3$	84. Answers will vary.
86. Answers will vary.

Attributions

This chapter has been adapted from “Divide Monomials” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.3 Integer Exponents and Scientific Notation

Learning Objectives

By the end of this section, you will be able to:

Use the definition of a negative exponent
Simplify expressions with integer exponents
Convert from decimal notation to scientific notation
Convert scientific notation to decimal form
Multiply and divide using scientific notation

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

Quotient Property for Exponents

If $a$ is a real number, $a not equal to 0$ , and $mathematical expression$ are whole numbers, then

What if we just subtract exponents regardless of which is larger?

Let’s consider $x to the 2 over x to the 5$ .

We subtract the exponent in the denominator from the exponent in the numerator.

$mathematical expression$

We can also simplify $x to the 2 over x to the 5$ by dividing out common factors:

Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.

This implies that $x to the -3=1 over x to the 3$ and it leads us to the definition of a negative exponent.

Negative Exponent

If $n$ is an integer and $a not equal to 0$ , then $a to the -n=1 over a to the n$ .

The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression $x to the -3$ , we will take one more step and write $1 over x to the 3$ . The answer is considered to be in simplest form when it has only positive exponents.

EXAMPLE 1

Simplify: a) $4 to the -2$ b) $10 to the -3$ .

Solution

a)	$4 to the -2$
Use the definition of a negative exponent, $a to the -n=1 over a to the n$ .	$1 over 4 to the 2$
Simplify.	$1 over 16$
b)	$10 to the -3$
Use the definition of a negative exponent, $a to the -n=1 over a to the n$ .	$1 over 10 to the 3$
Simplify.	$1 over 1000$

TRY IT 1.1

Simplify: a) $2 to the -3$ b) $10 to the -7$ .

Show answer

a) $1 over 8$ b) $1 over 10 to the 7$

TRY IT 1.2

Simplify: a) $3 to the -2$ b) $10 to the -4$ .

Show answer

a) $1 over 9$ b) $1 over 10,000$

In (Example 1) we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

	$mathematical expression$
Use the definition of a negative exponent, $a to the -n=1 over a to the n$ .	$1 over 1 over a to the n$
Simplify the complex fraction.	$1 times a to the n over 1$
Multiply.	$a to the n$

This leads to the Property of Negative Exponents.

Property of Negative Exponents

If $n$ is an integer and $a not equal to 0$ , then $1 over a to the -n=a to the n$ .

EXAMPLE 2

Simplify: a) $1 over y to the -4$ b) $1 over 3 to the -2$ .

Solution

a)	$1 over y to the -4$
Use the property of a negative exponent, $1 over a to the -n=a to the n$ .	$y to the 4$
b)	$1 over 3 to the -2$
Use the property of a negative exponent, $1 over a to the -n=a to the n$ .	$3 to the 2$
Simplify.	$9$

TRY IT 2.1

Simplify: a) $1 over p to the -8$ b) $1 over 4 to the -3$ .

Show answer

a) $p to the 8$ b) $64$

TRY IT 2.2

Simplify: a) $1 over q to the -7$ b) $1 over 2 to the -4$ .

Show answer

a) $q to the 7$ b) $16$

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

	$(3 over 4) to the -2$
Use the definition of a negative exponent, $a to the -n=1 over a to the n$ .	$1 over (3 over 4) to the 2$
Simplify the denominator.	$1 over 9 over 16$
Simplify the complex fraction.	$16 over 9$
But we know that $16 over 9$ is $(4 over 3) to the 2$ .
This tells us that:	$(3 over 4) to the -2=(4 over 3) to the 2$

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Exponent Property

If $mathematical expression$ are real numbers, $a not equal to 0,b not equal to 0$ , and $n$ is an integer, then $(a over b) to the -n=(b over a) to the n$ .

EXAMPLE 3

Simplify: a) $(5 over 7) to the -2$ b) $(-2x over y) to the -3$ .

Solution

a)	$(5 over 7) to the -2$
Use the Quotient to a Negative Exponent Property, $(a over b) to the -n=(b over a) to the n$ .
Take the reciprocal of the fraction and change the sign of the exponent.	$(7 over 5) to the 2$
Simplify.	$49 over 25$
b)	$(-2x over y) to the -3$
Use the Quotient to a Negative Exponent Property, $(a over b) to the -n=(b over a) to the n$ .
Take the reciprocal of the fraction and change the sign of the exponent.	$(-y over 2x) to the 3$
Simplify.	$-y to the 3 over 8x to the 3$

TRY IT 3.1

Simplify: a) $(2 over 3) to the -4$ b) $(-6m over n) to the -2$ .

Show answer

a) $81 over 16$ b) $n to the 2 over 36m to the 2$

TRY IT 3.2

Simplify: a) $(3 over 5) to the -3$ b) $(-a over 2b) to the -4$ .

Show answer

a) $125 over 27$ b) $16b to the 4 over a to the 4$

When simplifying an expression with exponents, we must be careful to correctly identify the base.

EXAMPLE 4

Simplify: a) $(-3) to the -2$ b) $-3 to the -2$ c) $(-1 over 3) to the -2$ d) $-(1 over 3) to the -2$ .

Solution

a) Here the exponent applies to the base $-3$ .	$(-3) to the -2$
Take the reciprocal of the base and change the sign of the exponent.	$1 over (-3) to the -2$
Simplify.	$1 over 9$
b) The expression $-3 to the -2$ means “find the opposite of $3 to the -2$ .” Here the exponent applies to the base $(-1 over 3) to the$ .	$-3 to the -2$
Rewrite as a product with $-1$ .	$-1 times 3 to the -2$
Take the reciprocal of the base and change the sign of the exponent.	$-1 times 1 over 3 to the 2$
Simplify.	$-1 over 9$
c) Here the exponent applies to the base $(-1 over 3) to the$ .	$(-1 over 3) to the -2$
Take the reciprocal of the base and change the sign of the exponent.	$(-3 over 1) to the 2$
Simplify.	$9$
d) The expression $-(1 over 3) to the -2$ means “find the opposite of $(1 over 3) to the -2$ .” Here the exponent applies to the base $(1 over 3)$ .
Rewrite as a product with $-1$ .	$-1 times (1 over 3) to the -2$
Take the reciprocal of the base and change the sign of the exponent.	$-1 times (3 over 1) to the 2$
Simplify.	$-9$

TRY IT 4.1

Simplify: a) $(-5) to the -2$ b) $-5 to the -2$ c) $(-1 over 5) to the -2$ d) $-(1 over 5) to the -2$ .

Show answer

a) $1 over 25$ b) $-1 over 25$ c) 25 d) $-25$

TRY IT 4.2

Simplify: a) $(-7) to the -2$ b) $-7 to the -2$ , c) $(-1 over 7) to the -2$ d) $-(1 over 7) to the -2$ .

Show answer

a) $1 over 49$ b) $-1 over 49$ c) 49 d) $-49$

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

EXAMPLE 5

Simplify: a) $4 times 2 to the -1$ b) $(4 times 2) to the -1$ .

Solution

a) Do exponents before multiplication.	$4 times 2 to the -1$
Use $a to the -n=1 over a to the n$ .	$4 times 1 over 2 to the 1$
Simplify.	$2$
b)	$(4 times 2) to the -1$
Simplify inside the parentheses first.	$(8) to the -1$
Use $a to the -n=1 over a to the n$ .	$1 over 8 to the 1$
Simplify.	$1 over 8$

TRY IT 5.1

Simplify: a) $6 times 3 to the -1$ b) $(6 times 3) to the -1$ .

Show answer

a) $2$ b) $1 over 18$

TRY IT 5.2

Simplify: a) $8 times 2 to the -2$ b) $(8 times 2) to the -2$ .

Show answer

a) 2 b) $1 over 16$

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.

EXAMPLE 6

Simplify: a) $x to the -6$ b) $(u to the 4) to the -3$ .

Solution

a)	$x to the -6$
Use the definition of a negative exponent $a to the -n=1 over a to the n$	$1 over x to the 6$
b)	$(u to the 4) to the -3$
Use the definition of a negative exponent $a to the -n=1 over a to the n$ .	$1 over (u to the 4) to the 3$
Simplify.	$1 over u to the 12$

TRY IT 6.1

Simplify: a) $y to the -7$ b) $(z to the 3) to the -5$ .

Show answer

a) $1 over y to the 7$ b) $1 over z to the 15$

TRY IT 6.2

Simplify: a) $p to the -9$ b) $(q to the 4) to the -6$ .

Show answer

a) $1 over p to the 9$ b) $1 over q to the 24$

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.

EXAMPLE 7

Simplify: a) $5y to the -1$ b) $(5y) to the -1$ c) $(-5y) to the -1$ .

Solution

a) Notice the exponent applies to just the base.	$5y to the -1$
Take the reciprocal of $y$ and change the sign of the exponent.	$5 times 1 over y to the 1$
Simplify.	$5 over y$
b) Her the parentheses make the exponent apply to the base.	$(5y) to the -1$
Take the reciprocal of $5y$ and change the sign of the exponent.	$1 over (5y) to the 1$
Simplify.	$1 over 5y$
c) The base here is $-5y$ .	$(-5y) to the -1$
Take the reciprocal of $-5y$ and change the sign of the exponent.	$1 over (-5y) to the 1$
Simplify.	$1 over -5y$
Use $a over -b=-a over b$	$-1 over 5y$

TRY IT 7.1

Simplify: a) $8p to the -1$ b) $(8p) to the -1$ c) $(-8p) to the -1$ .

Show answer

a) $8 over p$ b) $1 over 8p$ c) $-1 over 8p$

TRY IT 7.2

Simplify: a) $11q to the -1$ b) $(11q) to the -1$ $-(11q) to the -1$ c) $(-11q) to the -1$ .

Show answer

a) $1 over 11q$ b) $1 over 11q$ $-1 over 11q$ c) $-1 over 11q$

With negative exponents, the Quotient Rule needs only one form $a to the m over a to the n=a to the m-n$ , for $a not equal to 0$ . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Summary of Exponent Properties

If $mathematical expression$ are real numbers, and $mathematical expression$ are integers, then

$mathematical expression$

EXAMPLE 8

Simplify: a) $x to the -4 times x to the 6$ b) $y to the -6 times y to the 4$ c) $z to the -5 times z to the -3$ .

Solution

$x to the -4 times x to the 6$

Use the Product Property, $a to the m times a to the n=a to the m+n$ . $x to the -4+6$

Simplify $x to the 2$
$y to the -6 times y to the 4$

Notice the same bases, so add the exponents. $y to the -6+4$

Simplify. $y to the -2$

Use the definition of a negative exponent, $1 over a to the n$ . $1 over y to the 2$
$z to the -5 times z to the -3$

Add the exponents, since the bases are the same. $z to the -5-3$

Simplify. $z to the -8$

Take the reciprocal and change the sign of the exponent, using the definition of a negative exponent. $1 over z to the 8$

TRY IT 8.1

Simplify: a) $x to the -3 times x to the 7$ b) $y to the -7 times y to the 2$ c) $z to the -4 times z to the -5$ .

Show answer

a) $x to the 4$ b) $1 over y to the 5$ c) $1 over z to the 9$

TRY IT 8.2

Simplify: a) $a to the -1 times a to the 6$ b) $b to the -8 times b to the 4$ c) $c to the -8 times c to the -7$ .

Show answer

a) $a to the 5$ b) $1 over b to the 4$ c) $1 over c to the 15$

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

EXAMPLE 9

Simplify: $(m to the 4n to the -3)(m to the -5n to the -2)$ .

Solution

	$(m to the 4n to the -3)(m to the -5n to the -2)$
Use the Commutative Property to get like bases together.	$m to the 4m to the -5 times n to the -2n to the -3$
Add the exponents for each base.	$m to the -1 times n to the -5$
Take the reciprocals and change the signs of the exponents.	$1 over m to the 1 times 1 over n to the 5$
Simplify.	$1 over mn to the 5$

TRY IT 9.1

Simplify: $(p to the 6q to the -2)(p to the -9q to the -1)$ .

Show answer

$1 over p to the 3q to the 3$

TRY IT 9.2

Simplify: $(r to the 5s to the -3)(r to the -7s to the -5)$ .

Show answer

$1 over r to the 2s to the 8$

If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.

EXAMPLE 10

Simplify: $(2x to the -6y to the 8)(-5x to the 5y to the -3)$ .

Solution

	$(2x to the -6y to the 8)(-5x to the 5y to the -3)$
Rewrite with the like bases together.	$2(-5) times (x to the -6x to the 5) times (y to the 8y to the -3)$
Multiply the coefficients and add the exponents of each variable.	$-10 times x to the -1 times y to the 5$
Use the definition of a negative exponent, $a to the -n=1 over a to the n$ .	$-10 times 1 over x to the 1 times y to the 5$
Simplify.	$-10y to the 5 over x$

TRY IT 10.1

Simplify: $(3u to the -5v to the 7)(-4u to the 4v to the -2)$ .

Show answer

$-12v to the 5 over u$

TRY IT 10.2

Simplify: $(-6c to the -6d to the 4)(-5c to the -2d to the -1)$ .

Show answer

$30d to the 3 over c to the 8$

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

EXAMPLE 11

Simplify: $(6k to the 3) to the -2$ .

Solution

	$(6k to the 3) to the -2$
Use the product to a Power Property, $(ab) to the m=a to the mb to the m$ .	$(6) to the -2(k to the 3) to the -2$
Use the Power Property, $(a to the m) to the n=a to the m times n$ .	$6 to the -2k to the -6$
Use the Definition of a Negative Exponent, $a to the -n=1 over a to the n$ .	$1 over 6 to the 2 times 1 over k to the 6$
Simplify.	$1 over 36k to the 6$

TRY IT 11.1

Simplify: $(-4x to the 4) to the -2$ .

Show answer

$1 over 16x to the 8$

TRY IT 11.2

Simplify: $(2b to the 3) to the -4$ .

Show answer

$1 over 16b to the 12$

EXAMPLE 12

Simplify: $(5x to the -3) to the 2$ .

Solution

	$(5x to the -3) to the 2$
Use the Product to a Power Property, $(ab) to the m=a to the mb to the m$ .	$5 to the 2(x to the -3) to the 2$
Simplify and multiply the exponents of $x$ using the Power Property, $(a to the m) to the n=a to the m times n$ .	$25 times x to the -6$
Rewrite by using the Definition of a Negative Exponent, $a to the -n=1 over a to the n$ .	$25 times 1 over x to the 6$
Simplify.	$25 over x to the 6$

TRY IT 12.1

Simplify: $(8a to the -4) to the 2$ .

Show answer

$64 over a to the 8$

TRY IT 12.2

Simplify: $(2c to the -4) to the 3$ .

Show answer

$8 over c to the 12$

To simplify a fraction, we use the Quotient Property and subtract the exponents.

EXAMPLE 13

Simplify: $r to the 5 over r to the -4$ .

Solution

	$r to the 5 over r to the -4$
Use the Quotient Property, $a to the m over a to the n=a to the m-n$ .	$r to the 5-(-4)$
Simplify.	$r to the 9$

TRY IT 13.1

Simplify: $x to the 8 over x to the -3$ .

Show answer

$x to the 11$

TRY IT 13.2

Simplify: $y to the 8 over y to the -6$ .

Show answer

$y to the 13$

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and $0.004$ . We know that 4,000 means $mathematical expression$ and 0.004 means $mathematical expression$ .

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

$mathematical expression$

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

$mathematical expression$

It is customary in scientific notation to use as the $mathematical expression$ multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10

$mathematical expression$

EXAMPLE 14

How to Convert from Decimal Notation to Scientific Notation

Write in scientific notation: 37,000.

Solution

In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7. In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.

TRY IT 14.1

Write in scientific notation: $96,000$ .

Show answer

$mathematical expression$

TRY IT 14.2

Write in scientific notation: $48,300$ .

Show answer

$mathematical expression$

HOW TO: Convert from decimal notation to scientific notation

Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Count the number of decimal places, n, that the decimal point was moved.
Write the number as a product with a power of 10.
If the original number is:
- greater than 1, the power of 10 will be 10ⁿ.
- between 0 and 1, the power of 10 will be 10⁻ⁿ.
Check.

EXAMPLE 15

Write in scientific notation: $0.0052$ .

Solution

The original number, $0.0052$ , is between 0 and 1 so we will have a negative power of 10


Move the decimal point to get 5.2, a number between 1 and 10.
Count the number of decimal places the point was moved.
Write as a product with a power of 10.
Check.
$mathematical expression$
$mathematical expression$

TRY IT 15.1

Write in scientific notation: $0.0078$ .

Show answer

$mathematical expression$

TRY IT 15.2

Write in scientific notation: $0.0129$ .

Show answer

$mathematical expression$

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

$mathematical expression$

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

$mathematical expression$

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

EXAMPLE 16

How to Convert Scientific Notation to Decimal Form

Convert to decimal form: $mathematical expression$ .

Solution

In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.

TRY IT 16.1

Convert to decimal form: $mathematical expression$ .

Show answer

1,300

TRY IT 16.2

Convert to decimal form: $mathematical expression$ .

Show answer

92,500

The steps are summarized below.

HOW TO: Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

Determine the exponent, $n$ , on the factor 10.
Move the decimal places, adding zeros if needed.
- If the exponent is positive, move the decimal point $n$ places to the right.
- If the exponent is negative, move the decimal point $|n|$ places to the left.
Check.

EXAMPLE 17

Convert to decimal form: $mathematical expression$ .

Solution


Determine the exponent, n, on the factor 10.
Since the exponent is negative, move the decimal point 2 places to the left.
Add zeros as needed for placeholders.

TRY IT 17.1

Convert to decimal form: $mathematical expression$ .

Show answer

0.00012

TRY IT 17.2

Convert to decimal form: $mathematical expression$ .

Show answer

0.075

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

EXAMPLE 18

Multiply. Write answers in decimal form: $mathematical expression$ .

Solution

	$(4 times 10 to the 5)(2 times 10 to the -7)$
Use the Commutative Property to rearrange the factors.	$4 times 2 times 10 to the 5 times 10 to the -7$
Multiply.	$8 times 10 to the -2$
Change to decimal form by moving the decimal two places left.	$0.08$

TRY IT 18.1

Multiply $mathematical expression$ . Write answers in decimal form.

Show answer

0.06

TRY IT 18.2

Multiply $mathematical expression$ . Write answers in decimal form.

Show answer

0.009

EXAMPLE 19

Divide. Write answers in decimal form: $mathematical expression$ .

Solution

	$9 times 10 to the 3 over 3 times 10 to the -2$
Separate the factors, rewriting as the product of two fractions.	$9 over 3 times 10 to the 3 over 10 to the -2$
Divide.	$3 times 10 to the 5$
Change to decimal form by moving the decimal five places right.	$300,000$

TRY IT 19.1

Divide $mathematical expression$ . Write answers in decimal form.

Show answer

400,000

TRY IT 19.2

Divide $mathematical expression$ . Write answers in decimal form.

Show answer

20,000

Access these online resources for additional instruction and practice with integer exponents and scientific notation:

Key Concepts

Property of Negative Exponents
- If $n$ is a positive integer and $a not equal to 0$ , then $1 over a to the -n=a to the n$
Quotient to a Negative Exponent
- If $a,b$ are real numbers, $b not equal to 0$ and $n$ is an integer , then $(a over b) to the -n=(b over a) to the n$
To convert a decimal to scientific notation:
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n$ , that the decimal point was moved.
3. Write the number as a product with a power of 10. If the original number is:
  - greater than 1, the power of 10 will be $10 to the n$
  - between 0 and 1, the power of 10 will be $10 to the -n$
4. Check.
To convert scientific notation to decimal form:
1. Determine the exponent, $n$ , on the factor 10.
2. Move the decimal places, adding zeros if needed.
  - If the exponent is positive, move the decimal point $n$ places to the right.
  - If the exponent is negative, move the decimal point $|n|$ places to the left.
3. Check

Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

1. a) $3 to the -4$ b) $10 to the -2$	2. a) $4 to the -2$ b) $10 to the -3$
3. a) $2 to the -8$ b) $10 to the -2$	4. a) $5 to the -3$ b) $10 to the -5$
5. a) $1 over c to the -5$ b) $1 over 5 to the -2$	6. a) $1 over c to the -5$ b) $1 over 3 to the -2$
7. a) $1 over t to the -9$ b) $1 over 10 to the -4$	8. a) $1 over q to the -10$ b) $1 over 10 to the -3$
9. a) $(3 over 10) to the -2$ b) $(-2 over cd) to the -3$	10. a) $(5 over 8) to the -2$ b) $(-3m over n) to the -2$
11. a) $(7 over 2) to the -3$ b) $(-3 over xy to the 2) to the -3$	12. a) $(4 over 9) to the -3$ b) $(-u to the 2 over 2v) to the -5$
13. a) $(-7) to the -2$ b) $-7 to the -2$ c) $(-1 over 7) to the -2$ d) $-(1 over 7) to the -2$	14. a) $(-5) to the -2$ b) $-5 to the -2$ c) $(-1 over 5) to the -2$ d) $-(1 over 5) to the -2$
15. a) $-5 to the -3$ b) $(-1 over 5) to the -3$ c) $-(1 over 5) to the -3$ d) $(-5) to the -3$	16. a) $-3 to the -3$ b) $(-1 over 3) to the -3$ c) $(1 over 3) to the -3$ d) $(-3) to the -3$
17. a) $2 times 5 to the -1$ b) $(2 times 5) to the -1$	18. a) $3 times 5 to the -1$ b) $(3 times 5) to the -1$
19. a) $3 times 4 to the -2$ b) $(3 times 4) to the -2$	20. a) $4 times 5 to the -2$ b) $(4 times 5) to the -2$
21. a) $b to the -5$ b) $(k to the 2) to the -5$	22. a) $m to the -4$ b) $(x to the 3) to the -4$
23. a) $s to the -8$ b) $(a to the 9) to the -10$	24. a) $p to the -10$ b) $(q to the 6) to the -8$
25. a) $6r to the -1$ b) $(6r) to the -1$ c) $(-6r) to the -1$	26. a) $7n to the -1$ b) $(7n) to the -1$ c) $(-7n) to the -1$
27. a) $(2q) to the -4$ b) $2q to the -4$ c) $-2q to the -4$	28. a) $(3p) to the -2$ b) $3p to the -2$ c) $-3p to the -2$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

29. a) $s to the 3 times s to the -7$ b) $q to the -8 times q to the 3$ c) $y to the -2 times y to the -5$	30. a) $b to the 4b to the -8$ b) $r to the -2r to the 5$ c) $x to the -7x to the -3$
31. a) $y to the 5 times y to the -5$ b) $y times y to the 5$ c) $y times y to the -5$	32. a) $a to the 3 times a to the -3$ b) $a times a to the 3$ c) $a times a to the -3$
33. $x to the 4 times x to the -2 times x to the -3$	34. $(w to the 4x to the -5)(w to the -2x to the -4)$
35. $(m to the 3n to the -3)(m to the -5n to the -1)$	36. $(uv to the -2)(u to the -5v to the -3)$
37. $(pq to the -4)(p to the -6q to the -3)$	38. $(-6c to the -3d to the 9)(2c to the 4d to the -5)$
39. $(-2j to the -5k to the 8)(7j to the 2k to the -3)$	40. $(-4r to the -2s to the -8)(9r to the 4s to the 3)$
41. $(-5m to the 4n to the 6)(8m to the -5n to the -3)$	42. $(5x to the 2) to the -2$
43. $(4y to the 3) to the -3$	44. $(3z to the -3) to the 2$
45. $(2p to the -5) to the 2$	46. $t to the 9 over t to the -3$
47. $n to the 5 over n to the -2$	48. $x to the -7 over x to the -3$
49. $y to the -5 over y to the -10$

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

50. 57,000	51. 340,000
52. 8,750,000	53. 1,290,000
54. 0.026	55. 0.041
56. 0.00000871	57. 0.00000103

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

58. $mathematical expression$	59. $mathematical expression$
60. $mathematical expression$	61. $mathematical expression$
62. $mathematical expression$	63. $mathematical expression$
64. $mathematical expression$	65. $mathematical expression$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

66. $mathematical expression$	67. $mathematical expression$
68. $mathematical expression$	69. $mathematical expression$

In the following exercises, divide. Write your answer in decimal form.

70. $mathematical expression$	71. $mathematical expression$
72. $mathematical expression$	73. $mathematical expression$

Everyday Math

74. The population of the United States on July 1, 2010 was about 34,000,000. Write the number in scientific notation.	75. The population of the world on July 1, 2010 was more than 6,850,000,000. Write the number in scientific notation
76. The average width of a human hair is 0.0018 centimetres. Write the number in scientific notation.	77. The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.
78. In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $mathematical expression$ . Convert this number to decimal form.	79. At the start of 2012, the US federal budget had a deficit of more than $mathematical expression$ . Convert this number to decimal form.
80. The concentration of carbon dioxide in the atmosphere is $mathematical expression$ . Convert this number to decimal form.	81. The width of a proton is $mathematical expression$ of the width of an atom. Convert this number to decimal form.
82. Health care costs The Centers for Medicare and Medicaid projects that American consumers will spend more than $4 trillion on health care by 2017 Write 4 trillion in decimal notation. Write 4 trillion in scientific notation.	83. Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.
84. Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles. a) Write the number of miles in one light year in scientific notation. b)Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation.	85. Debt At the end of fiscal year 2019 the gross Canadian federal government debt was estimated to be approximately $685,450,000,000 ($685.45 billion), according to the Federal Budget. The population of Canada was approximately 37,590,000 people at the end of fiscal year 2019 a) Write the debt in scientific notation. b) Write the population in scientific notation. c) Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.

Writing Exercises.

86.

a) Explain the meaning of the exponent in the expression $2 to the 3$ .

b) Explain the meaning of the exponent in the expression $2 to the -3$ .

87. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answers

1. a) $1 over 81$ b) $1 over 100$	3. a) $1 over 256$ b) $1 over 100$	5. a) $c to the 5$ b) 25
7. a) $t to the 9$ b) 10000	9. a) $100 over 9$ b) $-c to the 3d to the 3 over 8$	11. a) $8 over 343$ b) $-x to the 3y to the 6 over 27$
13. a) $1 over 49$ b) $-1 over 49$ c) 49 d) $-49$	15. a) $-1 over 125$ b) $-125$ c) $-125$ d) $-1 over 125$	17. a) $2 over 5$ b) $1 over 10$
19. a) $3 over 16$ b) $1 over 144$	21. a) $1 over b to the 5$ b) $1 over k to the 10$	23. a) $1 over s to the 8$ b) $1 over a to the 90$
25. a) $6 over r$ b) $1 over 6r$ c) $-1 over 6r$	27. a) $1 over 16q to the 4$ b) $2 over q to the 4$ c) $-2 over q to the 4$	29. a) $1 over s to the 4$ b) $1 over q to the 5$ c) $1 over y to the 7$
31. a) 1 b) $y to the 6$ c) $1 over y to the 4$	33. $1 over x$	35. $1 over m to the 2n to the 4$
37. $1 over p to the 5q to the 7$	39. $-14k to the 5 over j to the 3$	41. $-40n to the 3 over m$
43. $1 over 64y to the 9$	45. $4 over p to the 10$	47. $n to the 7$
49. $y to the 5$	51. $mathematical expression$	53. $mathematical expression$
55. $mathematical expression$	57. $mathematical expression$	59. 830
61. 16,000,000,000	63. 0.038	65. 0.0000193
67. 0.02	69. $mathematical expression$	71. 500,000,000
73. 20,000,000	75. $mathematical expression$ .	77. $mathematical expression$
79. 15,000,000,000,000	81. 0.00001	83. $mathematical expression$
85. a) $mathematical expression$ b) $mathematical expression$ c) $mathematical expression$	87. Answers will vary

Attributions

This chapter has been adapted from “Integer Exponents and Scientific Notation” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.4 Simplify and Use Square Roots

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions with square roots
Estimate square roots
Approximate square roots
Simplify variable expressions with square roots
Use square roots in applications

Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number $n$ is multiplied by itself, we can write this as $n to the 2$ , which we read aloud as $mathematical expression$ For example, $8 to the 2$ is read as $mathematical expression$

We call $64$ the square of $8$ because $8 to the 2=64$ . Similarly, $121$ is the square of $11$ , because $11 to the 2=121$ .

Square of a Number

If $n to the 2=m$ , then $m$ is the square of $n$ .

Modeling Squares

Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.

A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.

This is why we say that the square of three is nine.

$3 to the 2=9$

The number $9$ is called a perfect square because it is the square of a whole number.

The chart shows the squares of the counting numbers $1$ through $15$ . You can refer to it to help you identify the perfect squares.

A table with two columns is shown. The first column is labeled “Number” and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled “Square” and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Perfect Squares

A perfect square is the square of a whole number.

What happens when you square a negative number?

$mathematical expression$

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from $-1$ to $-15$ .

Did you notice that these squares are the same as the squares of the positive numbers?

Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because $10 to the 2=100$ , we say $100$ is the square of $10$ . We can also say that $10$ is a square root of $100$ .

Square Root of a Number

A number whose square is $m$ is called a square root of $m$ .

If $n to the 2=m$ , then $n$ is a square root of $m$ .

Notice $(-10) to the 2=100$ also, so $-10$ is also a square root of $100$ . Therefore, both $10$ and $-10$ are square roots of $100$ .

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, $mathematical expression$ , stands for the positive square root. The positive square root is also called the principal square root.

Square Root Notation

$the square root of m$ is read as “the square root of $m.''$

$mathematical expression$ .

A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.

We can also use the radical sign for the square root of zero. Because $0 to the 2=0,the square root of 0=0$ . Notice that zero has only one square root.

The chart shows the square roots of the first $15$ perfect square numbers.

EXAMPLE 1

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Solution

a)
	$the square root of 25$
Since $5 to the 2=25$	$5$

b)
	$the square root of 121$
Since $11 to the 2=121$	$-11$

TRY IT 1.1

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

TRY IT 1.2

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

Every positive number has two square roots and the radical sign indicates the positive one. We write $the square root of 100=10$ . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $-the square root of 100=-10$ .

EXAMPLE 2

Simplify. a) $mathematical expression$ b) $mathematical expression$

Solution

a)
	$-the square root of 9$
The negative is in front of the radical sign.	$-3$

b)
	$-the square root of 144$
The negative is in front of the radical sign.	$-12$

TRY IT 2.1

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

−2
−15

TRY IT 2.2

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

−9
−8

Square Root of a Negative Number

Can we simplify $the square root of -25?$ Is there a number whose square is $-25?$

$mathematical expression$

None of the numbers that we have dealt with so far have a square that is $-25$ . Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to $the square root of -25$ . If we are asked to find the square root of any negative number, we say that the solution is not a real number.

EXAMPLE 3

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Solution

a) There is no real number whose square is $-169$ . Therefore, $the square root of -169$ is not a real number.

b) The negative is in front of the radical sign, so we find the opposite of the square root of $121$ .

	$-the square root of 121$
The negative is in front of the radical.	$-11$

TRY IT 3.1

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

not a real number
−9

TRY IT 3.2

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

−7
not a real number

Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

EXAMPLE 4

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Solution

a) Use the order of operations.
	$the square root of 25+the square root of 144$
Simplify each radical.	$5+12$
Add.	$17$

b) Use the order of operations.
	$the square root of 25+144$
Add under the radical sign.	$the square root of 169$
Simplify.	$13$

TRY IT 4.1

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

TRY IT 4.2

Simplify: a) $mathematical expression$ b) $mathematical expression$ .

Show answer

Notice the different answers in parts a) and b) of (Example 4). It is important to follow the order of operations correctly. In a), we took each square root first and then added them. In b), we added under the radical sign first and then found the square root.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled “Square root” and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.

We might conclude that the square roots of numbers between $4$ and $9$ will be between $2$ and $3$ , and they will not be whole numbers. Based on the pattern in the table above, we could say that $the square root of 5$ is between $2$ and $3$ . Using inequality symbols, we write

$2<the square root of 5<3$

EXAMPLE 5

Estimate $the square root of 60$ between two consecutive whole numbers.

Solution

Think of the perfect squares closest to $60$ . Make a small table of these perfect squares and their squares roots.

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled “Square root” and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.

$Locate 60 between two consecutive perfect squares.$	$49<60<64$
$mathematical expression$	$7<the square root of 60<8$

TRY IT 5.1

Estimate $the square root of 38$ between two consecutive whole numbers.

Show answer

$6<the square root of 38<7$

TRY IT 5.2

Estimate $the square root of 84$ between two consecutive whole numbers.

Show answer

$9<the square root of 84<10$

Approximate Square Roots with a Calculator

The square roots of numbers that are not perfect squares are not whole numbers, they are irrational numbers. Its decimal form does not stop and does not repeat. Are irrational numbers real numbers? Yes, they are. When we put together the irrational numbers and rational numbers, we get the set of real numbers.

Let’s see how we can use calculator to find the approximate square roots of those irrational numbers.

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the $mathematical expression$ or $the square root of x$ key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is $approximately$ and it is read approximately.

Suppose your calculator has a $10-digit$ display. Using it to find the square root of $5$ will give $2.236067977$ . This is the approximate square root of $5$ . When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

$the square root of 5 approximately 2.236067978$ . The square root of $5$ is the example of irrational number and its approximation displays nine digits after the decimal place.

You will seldom use this many digits for applications in algebra. So, if you wanted to round $the square root of 5$ to two decimal places, you would write

$the square root of 5 approximately 2.24$

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

$mathematical expression$

The squares are close, but not exactly equal, to $5$ .

EXAMPLE 6

Round $the square root of 17$ to two decimal places using a calculator.

Solution

	$the square root of 17$
Use the calculator square root key.	$4.123105626$
Round to two decimal places.	$4.12$
	$the square root of 17 approximately 4.12$

TRY IT 6.1

Round $the square root of 11$ to two decimal places.

Show answer

≈ 3.32

TRY IT 6.2

Round $the square root of 13$ to two decimal places.

Show answer

≈ 3.61

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider $the square root of 9x to the 2$ , where $x greater than or equal to 0$ . Can you think of an expression whose square is $9x to the 2?$

$mathematical expression$

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

EXAMPLE 7

Simplify: $the square root of x to the 2$ .

Solution

Think about what we would have to square to get $x to the 2$ . Algebraically, $(?) to the 2=x to the 2$

	$the square root of x to the 2$
Since $(x) to the 2=x to the 2$	$x$

TRY IT 7.1

Simplify: $the square root of y to the 2$ .

Show answer

TRY IT 7.2

Simplify: $the square root of m to the 2$ .

Show answer

EXAMPLE 8

Simplify: $the square root of 16x to the 2$ .

Solution

	$the square root of 16x to the 2$
$mathematical expression$	$4x$

TRY IT 8.1

Simplify: $the square root of 64x to the 2$ .

Show answer

TRY IT 8.2

Simplify: $the square root of 169y to the 2$ .

Show answer

13y

EXAMPLE 9

Simplify: $-the square root of 81y to the 2$ .

Solution

	$-the square root of 81y to the 2$
$mathematical expression$	$-9y$

TRY IT 9.1

Simplify: $-the square root of 121y to the 2$ .

Show answer

−11y

TRY IT 9.2

Simplify: $-the square root of 100p to the 2$ .

Show answer

−10p

EXAMPLE 10

Simplify: $the square root of 36x to the 2y to the 2$ .

Solution

	$the square root of 36x to the 2y to the 2$
$mathematical expression$	$6xy$

TRY IT 10.1

Simplify: $the square root of 100a to the 2b to the 2$ .

Show answer

10ab

TRY IT 10.2

Simplify: $the square root of 225m to the 2n to the 2$ .

Show answer

15mn

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

HOW TO: Use a strategy for applications with square roots.

Identify what you are asked to find.
Write a phrase that gives the information to find it.
Translate the phrase to an expression.
Simplify the expression.
Write a complete sentence that answers the question.

Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is $A$ square units, the length of a side is $the square root of A$ units. See the table below.

Area (square units)	Length of side (units)
$9$	$the square root of 9=3$
$144$	$the square root of 144=12$
$A$	$the square root of A$

EXAMPLE 11

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

Solution

We know the area of the square is $200$ square feet and want to find the length of the side. If the area of the square is $A$ square units, the length of a side is $the square root of A$ units.

What are you asked to find?	The length of each side of a square patio
Write a phrase.	The length of a side
Translate to an expression.	$the square root of A$
Evaluate $the square root of A$ when $A=200$ .	$the square root of 200$
Use your calculator.	$14.142135..$ .
Round to one decimal place.	$14.1 feet$
Write a sentence.	Each side of the patio should be $14.1$ feet.

TRY IT 11.1

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of $370$ square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?

Show answer

19.2 feet

TRY IT 11.2

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of $2704$ square centimetres. How long can a side of his mosaic be?

Show answer

52 centimetres

Square Roots and Gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of $h$ feet, the time in seconds it will take to reach the ground is found by evaluating the expression $the square root of h over 4$ . For example, if an object is dropped from a height of $64$ feet, we can find the time it takes to reach the ground by evaluating $the square root of 64 over 4$ .

	$the square root of 64 over 4$
Take the square root of 64.	$8 over 4$
Simplify the fraction.	$2$

It would take $2$ seconds for an object dropped from a height of $64$ feet to reach the ground.

EXAMPLE 12

Christy dropped her sunglasses from a bridge $400$ feet above a river. How many seconds does it take for the sunglasses to reach the river?

Solution

What are you asked to find?	The number of seconds it takes for the sunglasses to reach the river
Write a phrase.	The time it will take to reach the river
Translate to an expression.	$the square root of h over 4$
Evaluate $the square root of h over 4$ when $h=400$ .	$the square root of 400 over 4$
Find the square root of 400.	$20 over 4$
Simplify.	$5$
Write a sentence.	It will take 5 seconds for the sunglasses to reach the river.

TRY IT 12.1

A helicopter drops a rescue package from a height of $1296$ feet. How many seconds does it take for the package to reach the ground?

Show answer

9 seconds

TRY IT 12.2

A window washer drops a squeegee from a platform $196$ feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?

Show answer

3.5 seconds

Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is $d$ feet, then the speed of the car can be found by evaluating $the square root of 24d$ .

EXAMPLE 13

After a car accident, the skid marks for one car measured $190$ feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

Solution

What are you asked to find?	The speed of the car before the brakes were applied
Write a phrase.	The speed of the car
Translate to an expression.	$the square root of 24d$
Evaluate $mathematical expression$ when $mathematical expression$ .	$the square root of 24 times 190$
Multiply.	$the square root of 4,560$
Use your calculator.	$67.527772..$ .
Round to tenths.	$67.5$
Write a sentence.	The speed of the car was approximately 67.5 miles per hour.

TRY IT 13.1

An accident investigator measured the skid marks of a car and found their length was $76$ feet. To the nearest tenth, what was the speed of the car before the brakes were applied?

Show answer

42.7 mph

TRY IT 13.2

The skid marks of a vehicle involved in an accident were $122$ feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?

Show answer

54.1 mph

The Links to Literacy activity “Sea Squares” will provide you with another view of the topics covered in this section.

ACCESS ADDITIONAL ONLINE RESOURCES

Key Concepts

Square Root Notation $the square root of m$ is read ‘the square root of $m$ ’
If $m=n to the 2$ , then $the square root of m=n$ , for $n greater than or equal to 0$ .
Use a strategy for applications with square roots.
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Write a complete sentence that answers the question.

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

1. $the square root of 36$	2. $the square root of 4$
3. $the square root of 64$	4. $the square root of 144$
5. $-the square root of 4$	6. $-the square root of 100$
7. $-the square root of 1$	8. $-the square root of 121$
9. $the square root of -121$	10. $the square root of -36$
11. $the square root of -9$	12. $the square root of -49$
13. $the square root of 9+16$	14. $the square root of 25+144$
15. $the square root of 9+the square root of 16$	16. $the square root of 25+the square root of 144$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

17. $the square root of 70$	18. $the square root of 55$
19. $the square root of 200$	20. $the square root of 172$

Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

21. $the square root of 19$	22. $the square root of 21$
23. $the square root of 53$	24. $the square root of 47$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

25. $the square root of y to the 2$	26. $the square root of b to the 2$
27. $the square root of 49x to the 2$	28. $the square root of 100y to the 2$
29. $-the square root of 64a to the 2$	30. $-the square root of 25x to the 2$
31. $the square root of 144x to the 2y to the 2$	32. $the square root of 196a to the 2b to the 2$

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

33. Landscaping Reid wants to have a square garden plot in his backyard. He has enough compost to cover an area of $75$ square feet. How long can a side of his garden be?	34. Landscaping Tasha wants to make a square patio in her yard. She has enough concrete to pave an area of $130$ square feet. How long can a side of her patio be?
35. Gravity An airplane dropped a flare from a height of $1,024$ feet above a lake. How many seconds did it take for the flare to reach the water?	36. Gravity A hang glider dropped his cell phone from a height of $350$ feet. How many seconds did it take for the cell phone to reach the ground?
37. Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, $4,000$ feet above the Colorado River. How many seconds did it take for the hammer to reach the river?	38. Accident investigation The skid marks from a car involved in an accident measured $54$ feet. What was the speed of the car before the brakes were applied?
39. Accident investigation The skid marks from a car involved in an accident measured $216$ feet. What was the speed of the car before the brakes were applied?	40. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $175$ feet. What was the speed of the vehicle before the brakes were applied?
41. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was $117$ feet. What was the speed of the vehicle before the brakes were applied?

Everyday Math

42. Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy $625$ square centimetres of the designer tiles. How long can a side of the accent be?

43. Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for $2,025$ tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?

Writing Exercises

44. Why is there no real number equal to $the square root of -64?$

45. What is the difference between $9 to the 2$ and $the square root of 9?$

Answers

1. 6	3. 8	5. -2
7. -1	9. not a real number	11. not a real number
13. 5	15. 7	17. $8<the square root of 70<9$
19. $14<the square root of 200<15$	21. 4.36	23. 7.28
25. y	27. 7x	29. −8a
31. 12xy	33. 8.7 feet	35. 8 seconds
37. 15.8 seconds	39. 72 mph	41. 53.0 mph
43. 45 inches	45. Answers will vary. 9² reads: “nine squared” and means nine times itself. The expression $the square root of 9$ reads: “the square root of nine” which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.

Attributions

This chapter has been adapted from “Simplify and Use Square Roots” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.5 Simplify Square Roots

Learning Objectives

By the end of this section, you will be able to:

Use the Product Property to simplify square roots
Use the Quotient Property to simplify square roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that $the square root of 50$ is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use in (Simplify and Use Square Roots).

But what if we want to estimate $the square root of 500$ ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

Simplified Square Root

$the square root of a$ is considered simplified if $a$ has no perfect square factors.

So $the square root of 31$ is simplified. But $the square root of 32$ is not simplified, because 16 is a perfect square factor of 32

Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that $(ab) to the m=a to the mb to the m$ . The corresponding property of square roots says that $mathematical expression$ .

Product Property of Square Roots

If a, b are non-negative real numbers, then $mathematical expression$ .

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in (Example 1).

EXAMPLE 1

How To Use the Product Property to Simplify a Square Root

Simplify: $the square root of 50$ .

Solution

This figure has three columns and three rows. The first row says, “Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.” It then says, “25 is the largest perfect square factor of 50. 50 equals 25 times 2. Always write the perfect square factor first.” Then it shows the square root of 50 and the square root of 25 times 2. The second row says, “Step 2. Use the product rule to rewrite the radical as the product of two radicals.” The second column is empty, but the third column shows the square root of 25 times the square root of 2. The third row says, “Step 3. Simplify the square root of the perfect square.” The second column is empty, but the third column shows 5 times the square root of 2.

TRY IT 1.1

Simplify: $the square root of 48$ .

Show answer

$4the square root of 3$

TRY IT 1.2

Simplify: $the square root of 45$ .

Show answer

$3the square root of 5$

Notice in the previous example that the simplified form of $the square root of 50$ is $5the square root of 2$ , which is the product of an integer and a square root. We always write the integer in front of the square root.

HOW TO: Simplify a square root using the product property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

EXAMPLE 2

Simplify: $the square root of 500$ .

Solution

	$the square root of 500$
Rewrite the radicand as a product using the largest perfect square factor.	$the square root of 100 times 5$
Rewrite the radical as the product of two radicals.	$the square root of 100 times the square root of 5$
Simplify.	$10the square root of 5$

TRY IT 2.1

Simplify: $the square root of 288$ .

Show answer

$12the square root of 2$

TRY IT 2.2

Simplify: $the square root of 432$ .

Show answer

$12the square root of 3$

We could use the simplified form $10the square root of 5$ to estimate $the square root of 500$ . We know 5 is between 2 and 3, and $the square root of 500$ is $10the square root of 5$ . So $the square root of 500$ is between 20 and 30.

The next example is much like the previous examples, but with variables.

EXAMPLE 3

Simplify: $the square root of x to the 3$ .

Solution

	$the square root of x to the 3$
Rewrite the radicand as a product using the largest perfect square factor.	$the square root of x to the 2 times x$
Rewrite the radical as the product of two radicals.	$mathematical expression$
Simplify.	$xthe square root of x$

TRY IT 3.1

Simplify: $the square root of b to the 5$ .

Show answer

$b to the 2the square root of b$

TRY IT 3.2

Simplify: $the square root of p to the 9$ .

Show answer

$p to the 4the square root of p$

We follow the same procedure when there is a coefficient in the radical, too.

EXAMPLE 4

Simplify: $the square root of 25y to the 5.$

Solution

	$the square root of 25y to the 5$
Rewrite the radicand as a product using the largest perfect square factor.	$the square root of 25y to the 4 times y$
Rewrite the radical as the product of two radicals.	$mathematical expression$
Simplify.	$5y to the 2the square root of y$

TRY IT 4.1

Simplify: $the square root of 16x to the 7$ .

Show answer

$4x to the 3the square root of x$

TRY IT 4.2

Simplify: $the square root of 49v to the 9$ .

Show answer

$7v to the 4the square root of v$

In the next example both the constant and the variable have perfect square factors.

EXAMPLE 5

Simplify: $the square root of 72n to the 7$ .

Solution

	$the square root of 72n to the 7$
Rewrite the radicand as a product using the largest perfect square factor.	$the square root of 36n to the 6 times 2n$
Rewrite the radical as the product of two radicals.	$the square root of 36n to the 6 times the square root of 2n$
Simplify.	$6n to the 3the square root of 2n$

TRY IT 5.1

Simplify: $the square root of 32y to the 5$ .

Show answer

$4y to the 2the square root of 2y$

TRY IT 5.2

Simplify: $the square root of 75a to the 9$ .

Show answer

$5a to the 4the square root of 3a$

EXAMPLE 6

Simplify: $the square root of 63u to the 3v to the 5$ .

Solution

	$the square root of 63u to the 3v to the 5$
Rewrite the radicand as a product using the largest perfect square factor.	$the square root of 9u to the 2v to the 4 times 7uv$
Rewrite the radical as the product of two radicals.	$mathematical expression$
Simplify.	$3uv to the 2the square root of 7uv$

TRY IT 6.1

Simplify: $the square root of 98a to the 7b to the 5$ .

Show answer

$7a to the 3b to the 2the square root of 2ab to the$

TRY IT 6.2

Simplify: $the square root of 180m to the 9n to the 11$ .

Show answer

$6m to the 4n to the 5the square root of 5mn$

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify $the square root of 25+the square root of 144$ we must simplify each square root separately first, then add to get the sum of 17

The expression $the square root of 17+the square root of 7$ cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

EXAMPLE 7

Simplify: $3+the square root of 32$ .

Solution

	$3+the square root of 32$
Rewrite the radicand as a product using the largest perfect square factor.	$3+the square root of 16 times 2$
Rewrite the radical as the product of two radicals.	$mathematical expression$
Simplify.	$3+4the square root of 2$

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!

TRY IT 7.1

Simplify: $5+the square root of 75$ .

Show answer

$5+5the square root of 3$

TRY IT 7.2

Simplify: $2+the square root of 98$ .

Show answer

$2+7the square root of 2$

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

EXAMPLE 8

Simplify: $4-the square root of 48 over 2$ .

Solution

	$4-the square root of 48 over 2$
Rewrite the radicand as a product using the largest perfect square factor.	$4-the square root of 16 times 3 over 2$
Rewrite the radical as the product of two radicals.	$mathematical expression$
Simplify.	$4-4the square root of 3 over 2$
Factor the common factor from the numerator.	$4(1-the square root of 3) over 2$
Remove the common factor, 2, from the numerator and denominator.	$mathematical expression$
Simplify.	$2(1-the square root of 3)$

TRY IT 8.1

Simplify: $10-the square root of 75 over 5$ .

Show answer

$2-the square root of 3$

TRY IT 8.2

Simplify: $6-the square root of 45 over 3$ .

Show answer

$2-the square root of 5$

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

EXAMPLE 9

Simplify: $the square root of 9 over 64$ .

Solution

	$the square root of 9 over 64$
$Since(3 over 8) to the 2=9 over 64$	$3 over 8$

TRY IT 9.1

Simplify: $the square root of 25 over 16$ .

Show answer

$5 over 4$

TRY IT 9.2

Simplify: $the square root of 49 over 81$ .

Show answer

$7 over 9$

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

EXAMPLE 10

Simplify: $the square root of 45 over 80$ .

Solution

	$the square root of 45 over 80$
Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.	$the square root of 5 times 9 over 5 times 16$
Simplify the fraction by removing common factors.	$the square root of 9 over 16$
$Simplify(3 over 4) to the 2=9 over 16$	$3 over 4$

TRY IT 10.1

Simplify: $the square root of 75 over 48$ .

Show answer

$5 over 4$

TRY IT 10.2

Simplify: $the square root of 98 over 162$ .

Show answer

$7 over 9$

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, $a to the m over a to the n=a to the m-n,a not equal to 0$ .

EXAMPLE 11

Simplify: $the square root of m to the 6 over m to the 4$ .

Solution

	$the square root of m to the 6 over m to the 4$
Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents.	$the square root of m to the 2$
Simplify.	$m$

TRY IT 11.1

Simplify: $the square root of a to the 8 over a to the 6$ .

Show answer

$a$

TRY IT 11.2

Simplify: $the square root of x to the 14 over x to the 10$ .

Show answer

$x to the 2$

EXAMPLE 12

Simplify: $the square root of 48p to the 7 over 3p to the 3$ .

Solution

	$the square root of 48p to the 7 over 3p to the 3$
Simplify the fraction inside the radical first.	$the square root of 16p to the 4$
Simplify.	$4p to the 2$

TRY IT 12.1

Simplify: $the square root of 75x to the 5 over 3x$ .

Show answer

$5x to the 2$

TRY IT 12.2

Simplify: $the square root of 72z to the 12 over 2z to the 10$ .

Show answer

$6z$

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

$(a over b) to the m=a to the m over b to the m,b not equal to 0$

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Quotient Property of Square Roots

If a, b are non-negative real numbers and $b not equal to 0$ , then

$the square root of a over b=the square root of a over the square root of b$

EXAMPLE 13

Simplify: $the square root of 21 over 64$ .

Solution

	$the square root of 21 over 64$
We cannot simplify the fraction inside the radical. Rewrite using the quotient property.	$the square root of 21 over the square root of 64$
Simplify the square root of 64. The numerator cannot be simplified.	$the square root of 21 over 8$

TRY IT 13.1

Simplify: $the square root of 19 over 49$ .

Show answer

$the square root of 19 over 7$

TRY IT 13.2

Simplify: $the square root of 28 over 81$ .

Show answer

$2the square root of 7 over 9$

EXAMPLE 14

How to Use the Quotient Property to Simplify a Square Root

Simplify: $the square root of 27m to the 3 over 196$ .

Solution

$This table has three columns and three rows. The first row reads, “Step 1. Simplify the fraction in the radicand, if possible.” Then it shows that 27 m cubed over 196 cannot be simplified. Then it shows the square root of 27 m cubed over 196.$ The second row says, “Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.” Then it says, “We rewrite the square root of 27 m cubed over 196 as the quotient of the square root of 27 m cubed and the square root of 196.” Then it shows the square root of 27 m cubed over the square root of 196. The third row says, “Step 3. Simplify the radicals in the numerator and the denominator.” Then it says, “9 m squared and 196 are perfect squares.” It then shows the square root of 9 m squared time the square root of 3 m over the square root of 196. It then shows 3 m times the square root of 3 m over 14.

TRY IT 14.1

Simplify: $the square root of 24p to the 3 over 49$ .

Show answer

$2pthe square root of 6p over 7$

TRY IT 14.2

Simplify: $the square root of 48x to the 5 over 100$ .

Show answer

$2x to the 2the square root of 3x over 5$

HOW TO: Simplify a square root using the quotient property.

Simplify the fraction in the radicand, if possible.
Use the Quotient Property to rewrite the radical as the quotient of two radicals.
Simplify the radicals in the numerator and the denominator.

EXAMPLE 15

Simplify: $the square root of 45x to the 5 over y to the 4$ .

Solution

	$the square root of 45x to the 5 over y to the 4$
We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property.	$the square root of 45x to the 5 over the square root of y to the 4$
Simplify the radicals in the numerator and the denominator.	$mathematical expression$
Simplify.	$3x to the 2the square root of 5x over y to the 2$

TRY IT 15.1

Simplify: $the square root of 80m to the 3 over n to the 6$ .

Show answer

$4mthe square root of 5m over n to the 3$

TRY IT 15.2

Simplify: $the square root of 54u to the 7 over v to the 8$ .

Show answer

$3u to the 3the square root of 6u over v to the 4$

Be sure to simplify the fraction in the radicand first, if possible.

EXAMPLE 16

Simplify: $the square root of 81d to the 9 over 25d to the 4$ .

Solution

	$the square root of 81d to the 9 over 25d to the 4$
Simplify the fraction in the radicand.	$the square root of 81d to the 5 over 25$
Rewrite using the Quotient Property.	$the square root of 81d to the 5 over the square root of 25$
Simplify the radicals in the numerator and the denominator.	$mathematical expression$
Simplify.	$9d to the 2the square root of d over 5$

TRY IT 16.1

Simplify: $the square root of 64x to the 7 over 9x to the 3$ .

Show answer

$8x to the 2 over 3$

TRY IT 16.2

Simplify: $the square root of 16a to the 9 over 100a to the 5$ .

Show answer

$2a to the 2 over 5$

EXAMPLE 17

Simplify: $the square root of 18p to the 5q to the 7 over 32pq to the 2$ .

Solution

	$the square root of 18p to the 5q to the 7 over 32pq to the 2$
Simplify the fraction in the radicand, if possible.	$the square root of 9p to the 4q to the 5 over 16$
Rewrite using the Quotient Property.	$the square root of 9p to the 4q to the 5 over the square root of 16$
Simplify the radicals in the numerator and the denominator.	$mathematical expression$
Simplify.	$3p to the 2q to the 2the square root of q over 4$

TRY IT 17.1

Simplify: $the square root of 50x to the 5y to the 3 over 72x to the 4y$ .

Show answer

$5ythe square root of x over 6$

TRY IT 17.2

Simplify: $the square root of 48m to the 7n to the 2 over 125m to the 5n to the 9$ .

Show answer

$4mthe square root of 3 over 5n to the 3the square root of 5n$

Key Concepts

Simplified Square Root $the square root of a$ is considered simplified if $a$ has no perfect-square factors.
Product Property of Square Roots If a, b are non-negative real numbers, then
$mathematical expression$
Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the square root of the perfect square.
Quotient Property of Square Roots If a, b are non-negative real numbers and $b not equal to 0$ , then
$the square root of a over b=the square root of a over the square root of b$
Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
1. Simplify the fraction in the radicand, if possible.
2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
3. Simplify the radicals in the numerator and the denominator.

Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

1. $the square root of 27$	2. $the square root of 80$
3. $the square root of 125$	4. $the square root of 96$
5. $the square root of 200$	6. $the square root of 147$
7. $the square root of 450$	8. $the square root of 252$
9. $the square root of 800$	10. $the square root of 288$
11. $the square root of 675$	12. $the square root of 1250$
13. $the square root of x to the 7$	14. $the square root of y to the 11$
15. $the square root of p to the 3$	16. $the square root of q to the 5$
17. $the square root of m to the 13$	18. $the square root of n to the 21$
19. $the square root of r to the 25$	20. $the square root of s to the 33$
21. $the square root of 49n to the 17$	22. $the square root of 25m to the 9$
23. $the square root of 81r to the 15$	24. $the square root of 100s to the 19$
25. $the square root of 98m to the 5$	26. $the square root of 32n to the 11$
27. $the square root of 125r to the 13$	28. $the square root of 80s to the 15$
29. $the square root of 200p to the 13$	30. $the square root of 128q to the 3$
31. $the square root of 242m to the 23$	32. $the square root of 175n to the 13$
33. $the square root of 147m to the 7n to the 11$	34. $the square root of 48m to the 7n to the 5$
35. $the square root of 75r to the 13s to the 9$	36. $the square root of 96r to the 3s to the 3$
37. $the square root of 300p to the 9q to the 11$	38. $the square root of 192q to the 3r to the 7$
39. $the square root of 242m to the 13n to the 21$	40. $the square root of 150m to the 9n to the 3$
41. $5+the square root of 12$	42. $8+the square root of 96$
43. $1+the square root of 45$	44. $3+the square root of 125$
45. $10-the square root of 24 over 2$	46. $8-the square root of 80 over 4$
47. $3+the square root of 90 over 3$	48. $15+the square root of 75 over 5$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

49. $the square root of 49 over 64$	50. $the square root of 100 over 36$
51. $the square root of 121 over 16$	52. $the square root of 144 over 169$
53. $the square root of 72 over 98$	54. $the square root of 75 over 12$
55. $the square root of 9 over 25$	56. $the square root of 300 over 243$
57. $the square root of x to the 10 over x to the 6$	58. $the square root of p to the 20 over p to the 10$
59. $the square root of y to the 4 over y to the 8$	60. $the square root of q to the 8 over q to the 14$
61. $the square root of 200x to the 7 over 2x to the 3$	62. $the square root of 98y to the 11 over 2y to the 5$
63. $the square root of 96p to the 9 over 6p$	64. $the square root of 108q to the 10 over 3q to the 2$
65. $the square root of 36 over 35$	66. $the square root of 144 over 65$
67. $the square root of 20 over 81$	68. $the square root of 21 over 196$
69. $the square root of 96x to the 7 over 121$	70. $the square root of 108y to the 4 over 49$
71. $the square root of 300m to the 5 over 64$	72. $the square root of 125n to the 7 over 169$
73. $the square root of 98r to the 5 over 100$	74. $the square root of 180s to the 10 over 144$
75. $the square root of 28q to the 6 over 225$	76. $the square root of 150r to the 3 over 256$
77. $the square root of 75r to the 9 over s to the 8$	78. $the square root of 72x to the 5 over y to the 6$
79. $the square root of 28p to the 7 over q to the 2$	80. $the square root of 45r to the 3 over s to the 10$
81. $the square root of 100x to the 5 over 36x to the 3$	82. $the square root of 49r to the 12 over 16r to the 6$
83. $the square root of 121p to the 5 over 81p to the 2$	84. $the square root of 25r to the 8 over 64r$
85. $the square root of 32x to the 5y to the 3 over 18x to the 3y$	86. $the square root of 75r to the 6s to the 8 over 48rs to the 4$
87. $the square root of 27p to the 2q over 108p to the 5q to the 3$	88. $the square root of 50r to the 5s to the 2 over 128r to the 2s to the 5$

Everyday Math

89.

a) Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify $the square root of 288$ to determine the length and the width of his garden. Round to the nearest tenth of a foot.

b) Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify $the square root of 288-5$ to determine the length and width of the new garden. Round to the nearest tenth of a foot.

90.

a) Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify $the square root of 64 over 16$ to determine the number of seconds it takes for the sunglasses to reach the ground.

b) Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify $the square root of 144 over 16$ to determine the number of seconds it takes for the sunglasses to reach the ground.

Writing Exercises

91. Explain why $the square root of x to the 4=x to the 2$ . Then explain why $the square root of x to the 16=x to the 8$ .

92. Explain why $7+the square root of 9$ is not equal to $the square root of 7+9$ .

Answers

1. $3the square root of 3$	3. $5the square root of 5$	5. $10the square root of 2$
7. $15the square root of 2$	9. $20the square root of 2$	11. $15the square root of 3$
13. $x to the 3the square root of x$	15. $pthe square root of p$	17. $m to the 6the square root of m$
19. $r to the 12the square root of r$	21. $7n to the 8the square root of n$	23. $9r to the 7the square root of r$
25. $7m to the 2the square root of 2m$	27. $5r to the 6the square root of 5r$	29. $10p to the 6the square root of 2p$
31. $11m to the 11the square root of 2m$	33. $7m to the 3n to the 5the square root of 3mn$	35. $5r to the 6s to the 4the square root of 3rs$ 70)
37. $10p to the 4q to the 5the square root of 3pq$	39. $11m to the 6n to the 10the square root of 2mn$	41. $5+2the square root of 3$
43. $1+3the square root of 5$	45. $5-2the square root of 6$	47. $1+the square root of 10$
49. $7 over 8$	51. $11 over 4$	53. $6 over 7$
55. $3 over 5$	57. $x to the 2$	59. $1 over y to the 2$
61. $10x to the 2$	63. $4p to the 4$	65. $6 over the square root of 35$
67. $2the square root of 5 over 9$	69. $4x to the 3the square root of 6x over 11$	71. $10m to the 2the square root of 3m over 8$
73. $7r to the 2the square root of 2r over 10$	75. $2q to the 3the square root of 7 over 15$	77. $5r to the 4the square root of 3r over s to the 4$
79. $4p to the 3the square root of 7p over q$	81. $5x over 3$	83. $11pthe square root of p over 9$
85. $4xy over 3$	87. $1 over 2pqthe square root of p$	89. a) $mathematical expression$ b) $mathematical expression$
91. Answers will vary.

Attributions

This chapter has been adapted from “Simplify Square Roots” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

7.6 Chapter Review

Review Exercises

Simplify Expressions with Exponents

In the following exercises, simplify.

1. $17 to the 1$	2. $10 to the 4$
3. $(0.5) to the 3$	4. $(2 over 9) to the 2$
5. $-2 to the 6$	6. $(-2) to the 6$

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

7. $p to the 15 times p to the 16$	8. $x to the 4 times x to the 3$
9. $8 times 8 to the 5$	10. $4 to the 10 times 4 to the 6$
11. $y to the c times y to the 3$	12. $n times n to the 2 times n to the 4$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

13. $(5 to the 3) to the 2$	14. $(m to the 3) to the 5$
15. $(3 to the r) to the s$	16. $(y to the 4) to the x$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

17. $(-5y) to the 3$	18 $(4a) to the 2$
19. $(10xyz) to the 3$	20. $(2mn) to the 5$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

21. $(4a to the 3b to the 2) to the 3$	22. $(p to the 2) to the 5 times (p to the 3) to the 6$
23. $(2q to the 3) to the 4(3q) to the 2$	24. $(5x) to the 2(7x)$
25. $(2 over 5m to the 2n) to the 3$	26. $(1 over 3x to the 2) to the 2(1 over 2x) to the 3$

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

27. $10 to the 25 over 10 to the 5$	28. $u to the 24 over u to the 6$
29. $v to the 12 over v to the 48$	30. $3 to the 4 over 3 to the 6$
31. $5 over 5 to the 8$	32. $x over x to the 5$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

33. $x to the 0$	34. $75 to the 0$
35. $(-12 to the 0)$ $(-12) to the 0$	36. $-12 to the 0$
37. $(25x) to the 0$	38. $25x to the 0$
39. $(19n) to the 0-(25m) to the 0$	40. $19n to the 0-25m to the 0$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

41. $(m over 3) to the 4$	42. $(2 over 5) to the 3$
43. $(x over 2y) to the 6$	44. $(r over s) to the 8$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

45. $n to the 10 over (n to the 5) to the 2$	46. $(x to the 3) to the 5 over x to the 9$
47. $(r to the 8 over r to the 3) to the 4$	48. $(q to the 6 over q to the 8) to the 3$
49. $(3x to the 4 over 2y to the 2) to the 5$	50. $(c to the 2 over d to the 5) to the 9$
51. $(3n to the 2) to the 4(-5n to the 4) to the 3 over (-2n to the 5) to the 2$	52. $(v to the 3v to the 9 over v to the 6) to the 4$

Divide Monomials

In the following exercises, divide the monomials.

53. $64a to the 5b to the 9 over -16a to the 10b to the 3$	54. $-65y to the 14$ ÷ $5y to the 2$
55. $(8p to the 6q to the 2)(9p to the 3q to the 5) over 16p to the 8q to the 7$	56. $144x to the 15y to the 8z to the 3 over 18x to the 10y to the 2z to the 12$

Use the Definition of a Negative Exponent

In the following exercises, simplify.

57. $(-5) to the -3$	58. $9 to the -2$
59. $(6u) to the -3$	60. $3 times 4 to the -3$
61. $(3 over 4) to the -2$	62. $(2 over 5) to the -1$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

63. $q to the -6 times q to the -5$	64. $p to the -2 times p to the 8$
65. $(y to the 8) to the -1$	66. $(c to the -2d)(c to the -3d to the -2)$
67. $a to the 8 over a to the 12$	68. $(q to the -4) to the -3$
69. $r to the -2 over r to the -3$	70. $n to the 5 over n to the -4$

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

71. 0.00429	72. 8,500,000
73. In 2015, the population of the world was about 7,200,000,000 people.	74. The thickness of a dime is about 0.053 inches.

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

75. 1.5 × $10 to the 10$	76. 3.8 × $10 to the 5$
77. 5.5 × $10 to the -1$	78. 9.1 × $10 to the -7$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

79. 3.5 × $10 to the -2)$ (6.2 × $10 to the -1)$

80. 2 × $10 to the 5)$ (4 × $10 to the -3)$

In the following exercises, divide and write your answer in decimal form.

81. $mathematical expression$

82. $mathematical expression$

Simplify Expressions with Square Roots

In the following exercises, simplify.

83. $the square root of 144$

85. $-the square root of 81$

87. $the square root of -36$

89. $the square root of 64+225$

84. $the square root of 64$

86. $-the square root of 25$

88. $the square root of -9$

90. $the square root of 64+the square root of 225$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

91. $the square root of 155$

92. $the square root of 28$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

93. $the square root of 57$

94. $the square root of 15$

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

95. $the square root of 64b to the 2$

97. $the square root of 225m to the 2n to the 2$

99. $the square root of 49y to the 2$

101. $the square root of 121c to the 2d to the 2$

96. $the square root of q to the 2$

98. $-the square root of 121a to the 2$

100. $-the square root of 100q to the 2$

102. $the square root of 4a to the 2b to the 2$

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

103.Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of $30$ square feet. How long can a side of the flower garden be?

105. Accident investigation The skid marks of a car involved in an accident were $216$ feet. How fast had the car been going before applying the brakes?

104. Art Diego has $225$ square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be?

106. Gravity A hiker dropped a granola bar from a lookout spot $576$ feet above a valley. How long did it take the granola bar to reach the valley floor?

Review Exercise Answers

1. 17	3. 0.125
5. $-64$	7. $p to the 31$
9. $8 to the 6$	11. $y to the c+3$
13. $5 to the 6$	15. $3 to the rs$
17. $-125y to the 3$	19. $1000x to the 3y to the 3z to the 3$
21. $64a to the 9b to the 6$	23. $48q to the 14$
25. $8 over 125m to the 6n to the 3$	27. $10 to the 20$
29. $1 over v to the 36$	31. $1 over 5 to the 7$
33. 1	35. 1
37. 1	39. 0
41. $m to the 4 over 81$	43. $x to the 6 over 64y to the 6$
45. 1	47. $r to the 20$
49. $343x to the 20 over 32y to the 10$	51. $-10,125n to the 10 over 4$
53. $-4b to the 6 over a to the 5$	55. $9p over 2$
57. $-1 over 125$	59. $1 over 216u to the 3$
61. $16 over 9$	63. $1 over q to the 11$
65. $1 over y to the 8$	67. $1 over a to the 4$
69. $r$	71. $mathematical expression$
73. $mathematical expression$	75. $15,000,000,000$
77. $0.55$	79. $0.0217$
81. $0.0000003$	83. 12
85. −9	87. not a real number
89. 17	91. $12<the square root of 155<13$
93. 7.55	95. 8b
97. 15mn	99. 7y
101. 11cd	103. 5.5 feet
105. 72 mph

Practice Test

In the following exercises, simplify each expression.

1. $(-2 over 5) to the 3$	2. $u times u to the 4$
3. $(4a to the 3b to the 5) to the 2$	4. $n to the -2 over n to the -10$
5. $3 to the 8 over 3 to the 10$	6. $(v to the 2v to the 6 over v to the 4) to the 2$
7. $(87x to the 15y to the 3z to the 22) to the 0$	8. $(m to the 4 times m over m to the 3) to the 6$
9. $80c to the 8d to the 2 over 16cd to the 10$	10. $5 to the -2$
11. $q to the -4 times q to the -5$	12. $(4m) to the -3$
13. $mathematical expression$	14. $mathematical expression$
15. $the square root of 81$	16. $-the square root of 49$
17. $the square root of -16$	18. $the square root of b to the 2$
19. $-the square root of 64a to the 2$	20. $-the square root of 144q to the 2$
21. Convert 83,000,000 to scientific notation.	22. Convert $mathematical expression$ to decimal form.

Practice Test Answers

1. $-8 over 125$	2. $u to the 5$
3. $16a to the 6b to the 10$	4. $n to the 8$
5. $1 over 9$	6. $v to the 8$
7. $1$	8. $m to the 12$
9. $5c to the 7 over d to the 8$	10. $1 over 25$
11. $1 over q$	12. $1 over 64 m to the 3$
13. $2.1 times 10 to the -6$	14. $7.48 times 10 to the 4$
15. 9	16. -7
17. not a real number	18. $b$
19. $-8a$	20. $-12q$
21. $mathematical expression$	22. 0.0000691


Rewrite, a = a¹.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.

	$x to the -4 times x to the 6$
Use the Product Property, $a to the m times a to the n=a to the m+n$ .	$x to the -4+6$
Simplify	$x to the 2$

	$y to the -6 times y to the 4$
Notice the same bases, so add the exponents.	$y to the -6+4$
Simplify.	$y to the -2$
Use the definition of a negative exponent, $1 over a to the n$ .	$1 over y to the 2$

	$z to the -5 times z to the -3$
Add the exponents, since the bases are the same.	$z to the -5-3$
Simplify.	$z to the -8$
Take the reciprocal and change the sign of the exponent, using the definition of a negative exponent.	$1 over z to the 8$