To solve the expression [tex]\(\sqrt[5]{32^{-2}}\)[/tex], we can follow these steps:
1. Understand the Expression:
The expression [tex]\(\sqrt[5]{32^{-2}}\)[/tex] means we are taking the fifth root of [tex]\(32^{-2}\)[/tex].
2. Rewrite the Root as an Exponent:
Recall that the fifth root of a number [tex]\(x\)[/tex] can be written as [tex]\(x^{1/5}\)[/tex]. So, our expression becomes:
[tex]\[
(32^{-2})^{1/5}
\][/tex]
3. Simplify the Exponents:
When raising a power to another power, you multiply the exponents. Therefore:
[tex]\[
(32^{-2})^{1/5} = 32^{(-2) \cdot \frac{1}{5}}
\][/tex]
4. Multiply the Exponents:
Multiply [tex]\(-2\)[/tex] by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
(-2) \cdot \frac{1}{5} = -\frac{2}{5}
\][/tex]
So, the expression simplifies to:
[tex]\[
32^{-\frac{2}{5}}
\][/tex]
5. Interpret the Negative Exponent:
A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Therefore:
[tex]\[
32^{-\frac{2}{5}} = \frac{1}{32^{\frac{2}{5}}}
\][/tex]
6. Calculate [tex]\(32^{\frac{2}{5}}\)[/tex]:
This is equivalent to finding the fifth root of [tex]\(32\)[/tex] squared.
- First, find the fifth root of [tex]\(32\)[/tex]:
[tex]\[
\sqrt[5]{32} = 2
\][/tex]
because [tex]\(2^5 = 32\)[/tex].
- Next, square this result:
[tex]\[
2^2 = 4
\][/tex]
7. Substitute Back:
So, [tex]\(32^{\frac{2}{5}} = 4\)[/tex].
8. Find the Reciprocal:
Finally, taking the reciprocal gives:
[tex]\[
\frac{1}{4}
\][/tex]
Therefore, [tex]\(\sqrt[5]{32^{-2}} = \frac{1}{4}\)[/tex], which is numerically equal to 0.25. The result is approximately:
[tex]\[
0.24999999999999997
\][/tex]
