To solve the problem of finding the fifth root of [tex]\(\frac{1}{32}\)[/tex], follow these steps:
1. Understand the expression: We need to find [tex]\(\sqrt[5]{\frac{1}{32}}\)[/tex], which means the number that, when raised to the fifth power, equals [tex]\(\frac{1}{32}\)[/tex].
2. Rewrite the fraction: Notice that [tex]\(\frac{1}{32}\)[/tex] can be written as a power of 2. We know that [tex]\(32 = 2^5\)[/tex], so:
[tex]\[
\frac{1}{32} = 2^{-5}
\][/tex]
3. Apply the root: Now we need to find the fifth root of [tex]\(2^{-5}\)[/tex]. The fifth root can be represented as raising to the power of [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
\left(2^{-5}\right)^{\frac{1}{5}}
\][/tex]
4. Simplify the exponent: Use the properties of exponents to simplify:
[tex]\[
\left(2^{-5}\right)^{\frac{1}{5}} = 2^{-5 \times \frac{1}{5}} = 2^{-1}
\][/tex]
5. Evaluate the result: We know that [tex]\(2^{-1} = \frac{1}{2}\)[/tex], so:
[tex]\[
2^{-1} = \frac{1}{2}
\][/tex]
Therefore, the fifth root of [tex]\(\frac{1}{32}\)[/tex] is:
[tex]\[
\boxed{0.5}
\][/tex]
