Chapter 8: Q.511 (page 852)

In the following exercises, simplify
$ⓐ {(- 32)}^{\frac{1}{5}} ⓑ {(243)}^{- \frac{1}{5}} ⓒ - 125^{\frac{1}{3}}$

Short Answer

Expert verified

Simplifying,

$ⓐ {(- 32)}^{\frac{1}{5}} = - 2$

$ⓑ {(243)}^{- \frac{1}{5}} = \frac{1}{3}$

$ⓒ - 125^{\frac{1}{3}} = - 5$

Step by step solution

Step 1. Given information

We have given,

$ⓐ {(- 32)}^{\frac{1}{5}} ⓑ {(243)}^{- \frac{1}{5}} ⓒ - 125^{\frac{1}{3}}$

Step 2. Concept

Negative exponent rule:

$a^{- n} = \frac{1}{a^{n}}$

Power rule of the exponent:

${(a^{b})}^{c} = a^{b c}$

Part (a) Step 1. Explanation

We have,

${(- 32)}^{\frac{1}{5}}$

We will find the factors of 32,

$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^{5}$

Thus,

${(- 32)}^{\frac{1}{5}} = {(- 2^{5})}^{\frac{1}{5}}$

Using power rule of the exponents,

${(- 32)}^{\frac{1}{5}} = - 2$

Part (a) Step 2. Conclusion

Hence, value of ${(- 32)}^{\frac{1}{5}}$ is, -2.

Part (b) Step 1. Explanation

We have,

${(243)}^{- \frac{1}{5}}$

We will find the factors of 243,

$243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^{5}$

Thus,

${(243)}^{- \frac{1}{5}} = {(3^{5})}^{- \frac{1}{5}}$

Using negative exponent rule,

${(243)}^{- \frac{1}{5}} = \frac{1}{{(3^{5})}^{\frac{1}{5}}}$

Using power rule of the exponent,

${(243)}^{- \frac{1}{5}} = \frac{1}{3}$

Part (b) Step 2. Conclusion

Hence, value of ${(243)}^{- \frac{1}{5}}$ is, $\frac{1}{3}$ .

Part (c) Step 1. Explanation

We have,

${(- 125)}^{\frac{1}{3}}$

We will find the factors of 125,

$125 = 5 \times 5 \times 5 = 5^{3}$

Thus,

${(- 125)}^{\frac{1}{3}} = {(- 5^{3})}^{\frac{1}{3}}$

Using power rule of the exponent,

${(- 125)}^{\frac{1}{3}} = - 5$

Part (c) Step 2. Conclusion

Hence, value of ${(- 125)}^{\frac{1}{3}}$ is -5.

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In the following exercises, simplify
$ⓐ {(- 32)}^{\frac{1}{5}} ⓑ {(243)}^{- \frac{1}{5}} ⓒ - 125^{\frac{1}{3}}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Part (a) Step 1. Explanation

Part (a) Step 2. Conclusion

Part (b) Step 1. Explanation

Part (b) Step 2. Conclusion

Part (c) Step 1. Explanation

Part (c) Step 2. Conclusion

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In the following exercises, simplifyⓐ(−32)15ⓑ(243)-15ⓒ−12513

Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Part (a) Step 1. Explanation

Part (a) Step 2. Conclusion

Part (b) Step 1. Explanation

Part (b) Step 2. Conclusion

Part (c) Step 1. Explanation

Part (c) Step 2. Conclusion

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In the following exercises, simplify
$ⓐ {(- 32)}^{\frac{1}{5}} ⓑ {(243)}^{- \frac{1}{5}} ⓒ - 125^{\frac{1}{3}}$