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Chapter 3: Problem 5

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ 32^{-4 / 5} $$

Short Answer

Expert verified
The short answer is \(32^{-4 / 5} = \frac{1}{16}\). To reach this conclusion, we applied exponent rules step by step, first expressing the negative exponent as a reciprocal, then expressing the exponent as a root, evaluating the 5th root of 32, calculating the 4th power of the 5th root, and finally combining the results.

Step by step solution

01

Express negative exponent as a reciprocal

The first step is to understand that a negative exponent means taking the reciprocal of the base. So, \(-4/5\) would mean taking the 5th root of \(32\) and raising the result to the power of \(-4\). In other words: \[32^{-4 / 5} = \frac{1}{32^{4 / 5}}\]
02

Express the exponent as a root

Now, dealing with the positive exponent of \(4/5\), this means taking the 5th root of \(32\) (the denominator of the fraction) and raising the result to the 4th power (the numerator of the fraction): \[\frac{1}{32^{4 / 5}} = \frac{1}{(\sqrt[5]{32})^4}\]
03

Evaluate the 5th root of 32

The 5th root of \(32\) can be expressed as: \[\sqrt[5]{32} = 2\] This is because \(2^5 = 32\).
04

Calculate the 4th power of the 5th root

Now that we have evaluated the 5th root of \(32\), we need to raise it to the 4th power: \[(\sqrt[5]{32})^4 = 2^4\]
05

Evaluate the 4th power

Calculating the 4th power of \(2\) gives us: \[2^4 = 16\]
06

Combine the results

Now that we have calculated the 4th power of the 5th root, we can substitute it back into our original expression: \[\frac{1}{(\sqrt[5]{32})^4} = \frac{1}{16}\] So, our final result is: \[32^{-4 / 5} = \frac{1}{16}\]

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