To find the equivalent expression to [tex]\(\left(32^2 x^4 y^{12}\right)^{\frac{1}{5}}\)[/tex], let's simplify the given expression step by step.
1. Given Expression:
[tex]\[
\left(32^2 x^4 y^{12}\right)^{\frac{1}{5}}
\][/tex]
2. Simplify the Inside:
- Calculate [tex]\(32^2\)[/tex]:
[tex]\[
32^2 = 1024
\][/tex]
- The expression becomes:
[tex]\[
\left(1024 x^4 y^{12}\right)^{\frac{1}{5}}
\][/tex]
3. Apply the Exponent [tex]\(\frac{1}{5}\)[/tex] to Each Term:
- Rewrite the expression:
[tex]\[
\left(1024\right)^{\frac{1}{5}} \left(x^4\right)^{\frac{1}{5}} \left(y^{12}\right)^{\frac{1}{5}}
\][/tex]
4. Simplify Each Part:
- For [tex]\(1024^{\frac{1}{5}}\)[/tex]:
[tex]\[
1024^{\frac{1}{5}} = \left(2^{10}\right)^{\frac{1}{5}} = 2^{2} = 4
\][/tex]
- For [tex]\(\left(x^4\right)^{\frac{1}{5}}\)[/tex]:
[tex]\[
\left(x^4\right)^{\frac{1}{5}} = x^{\frac{4}{5}}
\][/tex]
- For [tex]\(\left(y^{12}\right)^{\frac{1}{5}}\)[/tex]:
[tex]\[
\left(y^{12}\right)^{\frac{1}{5}} = y^{\frac{12}{5}}
\][/tex]
5. Combine the Simplified Parts:
- The expression now looks like:
[tex]\[
4 \cdot x^{\frac{4}{5}} \cdot y^{\frac{12}{5}}
\][/tex]
6. Rewrite the Expression in a More Common Form:
- Notice that [tex]\( \frac{12}{5} = 2 + \frac{2}{5} \)[/tex]:
[tex]\[
y^{\frac{12}{5}} = y^2 \cdot y^{\frac{2}{5}}
\][/tex]
- The expression becomes:
[tex]\[
4 \cdot x^{\frac{4}{5}} \cdot y^2 \cdot y^{\frac{2}{5}} \Rightarrow 4 y^2 \cdot \left(x^{\frac{4}{5}} y^{\frac{2}{5}}\right)
\][/tex]
- Recognize a common term:
[tex]\[
x^{\frac{4}{5}} y^{\frac{2}{5}} = \left(x^4 y^2\right)^{\frac{1}{5}}
\][/tex]
Finally, the expression simplifies to:
[tex]\[
4 y^2 \sqrt[5]{x^4 y^2}
\][/tex]
So, the correct answer is:
[tex]\[
4 y^2 \sqrt[5]{x^4 y^2}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{4 y^2 \sqrt[5]{x^4 y^2}}
\][/tex]
