To find the expression equivalent to [tex]\(\left(256 x^{16}\right)^{\frac{1}{4}}\)[/tex], we need to handle the simplification step-by-step.
1. Start by rewriting the expression:
[tex]\[
\left(256 x^{16}\right)^{\frac{1}{4}}
\][/tex]
2. Apply the exponent [tex]\(\frac{1}{4}\)[/tex] to both the numerical part [tex]\(256\)[/tex] and the variable part [tex]\(x^{16}\)[/tex]:
[tex]\[
(256)^{\frac{1}{4}} \cdot \left(x^{16}\right)^{\frac{1}{4}}
\][/tex]
3. Simplify [tex]\(256^{\frac{1}{4}}\)[/tex]:
[tex]\[
256 = 4^4
\][/tex]
So,
[tex]\[
(256)^{\frac{1}{4}} = (4^4)^{\frac{1}{4}} = 4
\][/tex]
4. Simplify [tex]\(\left(x^{16}\right)^{\frac{1}{4}}\)[/tex]:
[tex]\[
\left(x^{16}\right)^{\frac{1}{4}} = x^{16 \cdot \frac{1}{4}} = x^4
\][/tex]
5. Combine the simplified parts:
[tex]\[
4 \cdot x^4 = 4x^4
\][/tex]
Therefore, the expression [tex]\(\left(256 x^{16}\right)^{\frac{1}{4}}\)[/tex] simplifies to [tex]\(\boxed{4x^4}\)[/tex].
