To determine the equivalent expression for [tex]\(\frac{6ab}{(a^0 b^2)^4}\)[/tex], we will follow a step-by-step approach to simplify the given expression.
1. Simplify the denominator:
- Recall that [tex]\(a^0 = 1\)[/tex] for any value of [tex]\(a\)[/tex]. This means that [tex]\(a^0\)[/tex] effectively disappears in the multiplication.
- Hence, inside the parenthesis, we have [tex]\(a^0 b^2 = 1 \cdot b^2 = b^2\)[/tex].
2. Raise the denominator to the power 4:
- We need to raise [tex]\(b^2\)[/tex] to the power 4:
[tex]\[
(b^2)^4 = b^{2 \cdot 4} = b^8
\][/tex]
3. Rewrite the original expression with the simplified denominator:
- Substitute [tex]\(b^8\)[/tex] into the denominator of the original expression:
[tex]\[
\frac{6ab}{b^8}
\][/tex]
4. Simplify the fraction:
- When simplifying the fraction involving exponents with the same base, subtract the exponents:
[tex]\[
\frac{6ab}{b^8} = 6a \cdot \frac{b}{b^8} = 6a \cdot b^{1-8} = 6a \cdot b^{-7} = \frac{6a}{b^7}
\][/tex]
Therefore, the equivalent expression for [tex]\(\frac{6ab}{(a^0 b^2)^4}\)[/tex] is [tex]\(\frac{6a}{b^7}\)[/tex].
Hence, the correct answer is:
C. [tex]\(\frac{6 a}{b^7}\)[/tex]
