Let's begin by simplifying the given expression step-by-step:
The given expression is:
[tex]\[
\frac{6ab}{\left(a^0 b^2\right)^4}
\][/tex]
We start by simplifying the denominator, [tex]\(\left(a^0 b^2\right)^4\)[/tex].
1. First, recall that any number or variable to the power of 0 is 1. Thus, [tex]\(a^0 = 1\)[/tex].
Therefore:
[tex]\[
\left(a^0 b^2\right)^4 = \left(1 \cdot b^2\right)^4 = (b^2)^4
\][/tex]
2. Next, simplify [tex]\( (b^2)^4 \)[/tex]:
When raising a power to another power, we multiply the exponents:
[tex]\[
(b^2)^4 = b^{2 \cdot 4} = b^8
\][/tex]
Thus, the original expression simplifies to:
[tex]\[
\frac{6ab}{b^8}
\][/tex]
3. Now, simplify the fraction [tex]\(\frac{6ab}{b^8}\)[/tex]:
When dividing terms 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]
So, [tex]\(\frac{6ab}{\left(a^0 b^2\right)^4}\)[/tex] simplifies to [tex]\(\frac{6a}{b^7}\)[/tex].
4. Compare the simplified expression with the given options:
A. [tex]\(\frac{6a}{b^5}\)[/tex]
B. [tex]\(\frac{6}{a^3 b^T}\)[/tex]
C. [tex]\(\frac{6a}{b^T}\)[/tex]
D. [tex]\(\frac{6}{a^8 b^5}\)[/tex]
None of the provided options [tex]\(\boxed{A}\)[/tex], [tex]\(\boxed{B}\)[/tex], [tex]\(\boxed{C}\)[/tex], or [tex]\(\boxed{D}\)[/tex] match the simplified expression [tex]\(\frac{6a}{b^7}\)[/tex].
Therefore, the correct answer is not listed among the provided choices.
