Which expression is equivalent to [tex]\left(256 x^{16}\right)^{\frac{1}{4}}[/tex]?

A. [tex]4 x^2[/tex]
B. [tex]4 x^4[/tex]
C. [tex]64 x^2[/tex]
D. [tex]64 x^4[/tex]



Answer :

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].