To determine which expression is equivalent to [tex]\(\left(256 x^{16}\right)^{\frac{1}{4}}\)[/tex], we should simplify the given expression step by step:
1. Starting with the expression:
[tex]\[
\left(256 x^{16}\right)^{\frac{1}{4}}
\][/tex]
2. Distribute the exponent [tex]\(\frac{1}{4}\)[/tex] to both terms inside the parentheses:
[tex]\[
\left(256\right)^{\frac{1}{4}} \left(x^{16}\right)^{\frac{1}{4}}
\][/tex]
3. Simplify [tex]\(\left(256\right)^{\frac{1}{4}}\)[/tex]:
- Recall that [tex]\(256\)[/tex] is [tex]\(2^8\)[/tex].
- Thus,
[tex]\[
(256)^{\frac{1}{4}} = (2^8)^{\frac{1}{4}} = 2^{8 \cdot \frac{1}{4}} = 2^2 = 4
\][/tex]
4. Simplify [tex]\(\left(x^{16}\right)^{\frac{1}{4}}\)[/tex]:
- Use the power rule for exponents [tex]\((a^m)^n = a^{mn}\)[/tex].
- Therefore,
[tex]\[
(x^{16})^{\frac{1}{4}} = x^{16 \cdot \frac{1}{4}} = x^4
\][/tex]
5. Combine the simplified parts:
[tex]\[
4 \cdot x^4
\][/tex]
Thus, the equivalent expression to [tex]\(\left(256 x^{16}\right)^{\frac{1}{4}}\)[/tex] is:
[tex]\[
4 x^4
\][/tex]
The correct answer is [tex]\(4 x^4\)[/tex].
