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1.3 Radicals and Rational Exponents

Simplifying square roots with the product and quotient rules, combining like radicals, rationalizing denominators, and rewriting roots as fractional exponents.

Algebra and Trigonometry 2e, 1.3 Radicals and Rational Exponents

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63 exercises

Evaluating Square Roots

Evaluate each expression.

  1. ⓐ 100 \sqrt{100}
  2. ⓑ 16 \sqrt{\sqrt{16}}
  3. ⓒ 25+144 \sqrt{25 + 144}
  4. ⓓ 49 − 81 \sqrt{49} - \sqrt{81}
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1.3 Exercise 9

Evaluate each expression.

  1. ⓐ 225 \sqrt{225}
  2. ⓑ 81 \sqrt{\sqrt{81}}
  3. ⓒ 25−9 \sqrt{25 - 9}
  4. ⓓ 36 + 121 \sqrt{36} + \sqrt{121}
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Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

  1. ⓐ 300 \sqrt{300}
  2. ⓑ 162 a 5 b 4 \sqrt{162 a^{5} b^{4}}
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1.3 Exercise 11

Simplify 50 x 2 y 3 z . \sqrt{50 x^{2} y^{3} z} .

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Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.
12 ⋅ 3 \sqrt{12} \cdot \sqrt{3}

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1.3 Exercise 13

Simplify 50x ⋅ 2x \sqrt{50 x} \cdot \sqrt{2 x} assuming x>0. x > 0.

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Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

5 36 \sqrt{\frac{5}{36}}

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1.3 Exercise 15

Simplify 2 x 2 9 y 4 . \sqrt{\frac{2 x^{2}}{9 y^{4}}} .

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Using the Quotient Rule to Simplify an Expression with Two Square Roots

Simplify the radical expression.

234 x 11 y 26 x 7 y \frac{\sqrt{234 x^{11} y}}{\sqrt{26 x^{7} y}}

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1.3 Exercise 17

Simplify 9 a 5 b 14 3 a 4 b 5 . \frac{\sqrt{9 a^{5} b^{14}}}{\sqrt{3 a^{4} b^{5}}} .

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Adding Square Roots

Add 5 12 +2 3 . 5 \sqrt{12} + 2 \sqrt{3} .

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1.3 Exercise 19

Add 5 +6 20 . \sqrt{5} + 6 \sqrt{20} .

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Subtracting Square Roots

Subtract 20 72 a 3 b 4 c −14 8 a 3 b 4 c . 20 \sqrt{72 a^{3} b^{4} c} - 14 \sqrt{8 a^{3} b^{4} c} .

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1.3 Exercise 21

Subtract 3 80x −4 45x . 3 \sqrt{80 x} - 4 \sqrt{45 x} .

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Rationalizing a Denominator Containing a Single Term

Write 2 3 3 10 \frac{2 \sqrt{3}}{3 \sqrt{10}} in simplest form.

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1.3 Exercise 23

Write 12 3 2 \frac{12 \sqrt{3}}{\sqrt{2}} in simplest form.

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Rationalizing a Denominator Containing Two Terms

Write 4 1+ 5 \frac{4}{1 + \sqrt{5}} in simplest form.

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1.3 Exercise 25

Write 7 2+ 3 \frac{7}{2 + \sqrt{3}} in simplest form.

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Simplifying n th Roots

Simplify each of the following:

  1. ⓐ −32 5 \sqrt[5]{−32}
  2. ⓑ 4 4 ⋅ 1,024 4 \sqrt[4]{4} \cdot \sqrt[4]{1 , 024}
  3. ⓒ − 8 x 6 125 3 - \sqrt[3]{\frac{8 x^{6}}{125}}
  4. ⓓ 8 3 4 − 48 4 8 \sqrt[4]{3} - \sqrt[4]{48}
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1.3 Exercise 27

Simplify.

  1. ⓐ −216 3 \sqrt[3]{−216}
  2. ⓑ 3 80 4 5 4 \frac{3 \sqrt[4]{80}}{\sqrt[4]{5}}
  3. ⓒ 6 9,000 3 +7 576 3 6 \sqrt[3]{9 , 000} + 7 \sqrt[3]{576}
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Writing Rational Exponents as Radicals

Write 343 2 3 343^{\frac{2}{3}} as a radical. Simplify.

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1.3 Exercise 29

Write 9 5 2 9^{\frac{5}{2}} as a radical. Simplify.

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Writing Radicals as Rational Exponents

Write 4 a 2 7 \frac{4}{\sqrt[7]{a^{2}}} using a rational exponent.

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1.3 Exercise 31

Write x (5y) 9 x \sqrt{\left(5 y\right)^{9}} using a rational exponent.

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Simplifying Rational Exponents

Simplify:

  1. ⓐ 5( 2 x 3 4 )( 3 x 1 5 ) 5 \left(2 x^{\frac{3}{4}}\right) \left(3 x^{\frac{1}{5}}\right)
  2. ⓑ ( 16 9 ) − 1 2 \left(\frac{16}{9}\right)^{- \frac{1}{2}}
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1.3 Exercise 33

Simplify ( 8x ) 1 3 ( 14 x 6 5 ). \left(8 x\right)^{\frac{1}{3}} \left(14 x^{\frac{6}{5}}\right) .

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1.3 Exercise 34

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

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1.3 Exercise 36

Every number will have two square roots. What is the principal square root?

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1.3 Exercise 38

Simplify each expression.

256 \sqrt{256}

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1.3 Exercise 40

Simplify each expression.

4( 9+16 ) \sqrt{4 \left(9 + 16\right)}

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1.3 Exercise 42

Simplify each expression.

196 \sqrt{196}

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1.3 Exercise 44

Simplify each expression.

98 \sqrt{98}

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1.3 Exercise 46

Simplify each expression.

81 5 \sqrt{\frac{81}{5}}

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1.3 Exercise 48

Simplify each expression.

169 + 144 \sqrt{169} + \sqrt{144}

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1.3 Exercise 50

Simplify each expression.

18 162 \frac{18}{\sqrt{162}}

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1.3 Exercise 52

Simplify each expression.

14 6 −6 24 14 \sqrt{6} - 6 \sqrt{24}

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1.3 Exercise 54

Simplify each expression.

150 \sqrt{150}

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1.3 Exercise 56

Simplify each expression.

( 42 )( 30 ) \left(\sqrt{42}\right) \left(\sqrt{30}\right)

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1.3 Exercise 58

Simplify each expression.

4 225 \sqrt{\frac{4}{225}}

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1.3 Exercise 60

Simplify each expression.

360 361 \sqrt{\frac{360}{361}}

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1.3 Exercise 62

Simplify each expression.

8 1− 17 \frac{8}{1 - \sqrt{17}}

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1.3 Exercise 64

Simplify each expression.

128 3 +3 2 3 \sqrt[3]{128} + 3 \sqrt[3]{2}

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1.3 Exercise 66

Simplify each expression.

15 125 4 5 4 \frac{15 \sqrt[4]{125}}{\sqrt[4]{5}}

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1.3 Exercise 68

Simplify each expression.

400 x 4 \sqrt{400 x^{4}}

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1.3 Exercise 70

Simplify each expression.

49p \sqrt{49 p}

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1.3 Exercise 72

Simplify each expression.

m 5 2 289 m^{\frac{5}{2}} \sqrt{289}

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1.3 Exercise 74

Simplify each expression.

3 a b 2 −b a 3 \sqrt{a b^{2}} - b \sqrt{a}

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1.3 Exercise 76

Simplify each expression.

225 x 3 49x \sqrt{\frac{225 x^{3}}{49 x}}

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1.3 Exercise 78

Simplify each expression.

50 y 8 \sqrt{50 y^{8}}

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1.3 Exercise 80

Simplify each expression.

32 14d \sqrt{\frac{32}{14 d}}

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1.3 Exercise 82

Simplify each expression.

8 1− 3x \frac{\sqrt{8}}{1 - \sqrt{3 x}}

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1.3 Exercise 84

Simplify each expression.

w 3 2 32 − w 3 2 50 w^{\frac{3}{2}} \sqrt{32} - w^{\frac{3}{2}} \sqrt{50}

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1.3 Exercise 86

Simplify each expression.

12x 2+2 3 \frac{\sqrt{12 x}}{2 + 2 \sqrt{3}}

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1.3 Exercise 88

Simplify each expression.

125 n 10 \sqrt{125 n^{10}}

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1.3 Exercise 90

Simplify each expression.

81m 361 m 2 \sqrt{\frac{81 m}{361 m^{2}}}

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1.3 Exercise 92

Simplify each expression.

144 324 d 2 \sqrt{\frac{144}{324 d^{2}}}

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1.3 Exercise 94

Simplify each expression.

162 x 6 16 x 4 4 \sqrt[4]{\frac{162 x^{6}}{16 x^{4}}}

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1.3 Exercise 96

Simplify each expression.

128 z 3 3 − −16 z 3 3 \sqrt[3]{128 z^{3}} - \sqrt[3]{−16 z^{3}}

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1.3 Exercise 98

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by eval…

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1.3 Exercise 100

Simplify each expression.

8 − 16 4− 2 − 2 1 2 \frac{\sqrt{8} - \sqrt{16}}{4 - \sqrt{2}} - 2^{\frac{1}{2}}

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1.3 Exercise 102

Simplify each expression.

m n 3 a 2 c −3 ⋅ a −7 n −2 m 2 c 4 \frac{\sqrt{m n^{3}}}{a^{2} \sqrt{c^{−3}}} \cdot \frac{a^{−7} n^{−2}}{\sqrt{m^{2} c^{4}}}

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1.3 Exercise 104

Simplify each expression.

x 64y +4 y 128y \frac{x \sqrt{64 y} + 4 \sqrt{y}}{\sqrt{128 y}}

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1.3 Exercise 106

Simplify each expression.

64 3 + 256 4 64 + 256 \sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64} + \sqrt{256}}}

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.