Simplify Square Roots

9.2 Simplify Square RootsLearning ObjectivesBy the end of this section, you will be able to:

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

Be Prepared!

Before you get started take this readiness quiz.

1. Simplify: 80176 .

If you missed this problem, review Example 1.65.

2. Simplify: n9

n3 .


3. Simplify:q4

q12 .


In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that50 is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use Figure 9.2.

But what if we want to estimate 500 ? If we simplify the square root first, we’ll be able to estimate it easily. There areother 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

a is considered simplified if a has no perfect square factors.

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

Use the Product Property to Simplify Square RootsThe properties we will use to simplify expressions with square roots are similar to the properties of exponents. We knowthat (ab)m = am bm . The corresponding property of square roots says that ab = a · b .

Product Property of Square Roots

If a, b are non-negative real numbers, then ab = a · b .

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to dothis in Example 9.12.

EXAMPLE 9.12 HOW TO USE THE PRODUCT PROPERTY TO SIMPLIFY A SQUARE ROOT

Simplify: 50 .

Solution

Chapter 9 Roots and Radicals 1023

TRY IT : : 9.23 Simplify: 48 .


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

EXAMPLE 9.13

Simplify: 500 .

Solution500

Rewrite the radicand as a product using thelargest perfect square factor.

100 · 5

Rewrite the radical as the product of tworadicals.

100 · 5

Simplify. 10 5



We could use the simplified form 10 5 to estimate 500 . We know 5 is between 2 and 3, and 500 is 10 5 . So 500is between 20 and 30.The next example is much like the previous examples, but with variables.

EXAMPLE 9.14

Simplify: x3 .

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 usingthe 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.

Step 1.

Step 2.Step 3.

1024 Chapter 9 Roots and Radicals

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

Solution

x3


x2 · x


x2 · x

Simplify. x x

TRY IT : : 9.27 Simplify: b5 .

TRY IT : : 9.28 Simplify: p9 .

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

EXAMPLE 9.15

Simplify: 25y5.

Solution

25y5


25y4 · y


25y4 · y

Simplify. 5y2 y

TRY IT : : 9.29 Simplify: 16x7 .

TRY IT : : 9.30 Simplify: 49v9 .

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

EXAMPLE 9.16

Simplify: 72n7 .

Solution

72n7


36n6 · 2n


36n6 · 2n

Simplify. 6n3 2n

TRY IT : : 9.31 Simplify: 32y5 .


TRY IT : : 9.32 Simplify: 75a9 .

EXAMPLE 9.17

Simplify: 63u3 v5 .

Solution

63u3 v5


9u2 v4 · 7uv


9u2 v4 · 7uv

Simplify. 3uv2 7uv

TRY IT : : 9.33 Simplify: 98a7 b5 .

TRY IT : : 9.34 Simplify: 180m9 n11 .

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

The expression 17 + 7 cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7contains 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 theresulting expression to the integer.

EXAMPLE 9.18

Simplify: 3 + 32 .

Solution3 + 32


3 + 16 · 2


3 + 16 · 2

Simplify. 3 + 4 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 integerand a variable—they are not like terms!

TRY IT : : 9.35 Simplify: 5 + 75 .

TRY IT : : 9.36 Simplify: 2 + 98 .

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

EXAMPLE 9.19



Simplify: 4 − 482 .

Solution4 − 48

2


4 − 16 · 32


4 − 16 · 32

Simplify. 4 − 4 32

Factor the common factor from thenumerator.

4⎛⎝1 − 3⎞

⎠

2

Remove the common factor, 2, from thenumerator and denominator.

2 · 2⎛⎝1 − 3⎞

⎠

2

Simplify. 2⎛⎝1 − 3⎞

⎠

TRY IT : : 9.37Simplify: 10 − 75

5 .

TRY IT : : 9.38Simplify: 6 − 45

3 .

Use the Quotient Property to Simplify Square RootsWhenever you have to simplify a square root, the first step you should take is to determine whether the radicand isa perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfectsquares.

EXAMPLE 9.20

Simplify: 964 .

Solution964

Since ⎛⎝38

⎞⎠

2= 9

6438

TRY IT : : 9.39Simplify: 25

16 .


81 .

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


EXAMPLE 9.21

Simplify: 4580 .

Solution4580

Simplify inside the radical fir t. Rewriteshowing the common factors of thenumerator and denominator.

5 · 95 · 16

Simplify the fraction by removing commonfactors.

916

Simplify. ⎛⎝34

⎞⎠2

= 916

34


48 .


162 .

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the nextexample we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their

exponents, am

an = am − n, a ≠ 0 .

EXAMPLE 9.22

Simplify: m6

m4 .

Solution

m6

m4

Simplify the fraction inside the radical fir t.

Divide the like bases by subtracting theexponents.

m2

Simplify. m

TRY IT : : 9.43Simplify: a8

a6 .

TRY IT : : 9.44Simplify: x14

x10 .

EXAMPLE 9.23



Simplify: 48p7

3p3 .

Solution

48p7

3p3

Simplify the fraction inside the radical fir t. 16p4

Simplify. 4p2

TRY IT : : 9.45Simplify: 75x5

3x .

TRY IT : : 9.46Simplify: 72z12

2z10 .

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

⎛⎝ab

⎞⎠

m= am

bm, b ≠ 0

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

Quotient Property of Square Roots

If a, b are non-negative real numbers and b ≠ 0 , then

ab = a

b

EXAMPLE 9.24

Simplify: 2164 .

Solution2164

We cannot simplify the fraction inside theradical. Rewrite using the quotientproperty.

2164

Simplify the square root of 64. Thenumerator cannot be simplified

218


49 .


81 .


EXAMPLE 9.25 HOW TO USE THE QUOTIENT PROPERTY TO SIMPLIFY A SQUARE ROOT

Simplify: 27m3

196 .

Solution

TRY IT : : 9.49Simplify: 24p3

49 .


100 .

EXAMPLE 9.26

Simplify: 45x5

y4 .

Solution

45x5

y4

We cannot simplify the fraction in theradicand. Rewrite using the QuotientProperty.

45x5

y4

Simplify the radicals in the numerator andthe denominator.

9x4 · 5xy2

Simplify. 3x2 5xy2

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.

Step 1.Step 2.Step 3.



TRY IT : : 9.51Simplify: 80m3

n6 .

TRY IT : : 9.52Simplify: 54u7

v8 .

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

EXAMPLE 9.27

Simplify: 81d9

25d4 .

Solution

81d9

25d4

Simplify the fraction in the radicand. 81d5

25

Rewrite using the Quotient Property. 81d5

25Simplify the radicals in the numerator andthe denominator.

81d4 · d5

Simplify. 9d2 d5


9x3 .

TRY IT : : 9.54Simplify: 16a9

100a5 .

EXAMPLE 9.28

Simplify:18p5 q7

32pq2 .

Solution

18p5 q7

32pq2

Simplify the fraction in the radicand, ifpossible.

9p4 q5

16

Rewrite using the Quotient Property. 9p4 q5

16Simplify the radicals in the numerator andthe denominator.

9p4 q4 · q4

Simplify. 3p2 q2 q4


Practice Makes Perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

53. 27 54. 80 55. 125

56. 96 57. 200 58. 147

59. 450 60. 252 61. 800

62. 288 63. 675 64. 1250

65. x7 66. y11 67. p3

68. q5 69. m13 70. n21

71. r25 72. s33 73. 49n17

74. 25m9 75. 81r15 76. 100s19

77. 98m5 78. 32n11 79. 125r13

80. 80s15 81. 200p13 82. 128q3

83. 242m23 84. 175n13 85. 147m7 n11

86. 48m7 n5 87. 75r13 s9 88. 96r3 s3

89. 300p9 q11 90. 192q3 r7 91. 242m13 n21

92. 150m9 n3 93. 5 + 12 94. 8 + 96

95. 1 + 45 96. 3 + 125 97. 10 − 242

98. 8 − 804 99. 3 + 90

3 100. 15 + 755

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

101. 4964 102. 100

36 103. 12116

104. 144169 105. 72

98 106. 7512

9.2 EXERCISES


107. 45125 108. 300

243 109. x10

x6

110.p20

p10 111.y4

y8 112.q8

q14

113. 200x7

2x3 114.98y11

2y5 115.96p9

6p

116.108q10

3q2117. 36

35 118. 14465

119. 2081 120. 21

196 121. 96x7

121

122.108y4

49123. 300m5

64 124. 125n7

169

125. 98r5

100 126. 180s10

144 127.28q6

225

128. 150r3

256 129. 75r9

s8 130. 72x5

y6

131.28p7

q2132. 45r3

s10 133. 100x5

36x3

134. 49r12

16r6 135.121p5

81p2136. 25r8

64r

137.32x5 y3

18x3 y138. 75r6 s8

48rs4 139.27p2 q

108p5 q3

140. 50r5 s2

128r2 s5



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Simplify Square Roots