Let's start with the given expression:
[tex]\[
\frac{6ab}{\left(a b^2\right)^2}
\][/tex]
First, we need to simplify the denominator [tex]\(\left(a b^2\right)^2\)[/tex]. Using the properties of exponents, we can rewrite this as:
[tex]\[
\left(a b^2\right)^2 = a^2 (b^2)^2 = a^2 b^4
\][/tex]
So, the expression becomes:
[tex]\[
\frac{6ab}{a^2 b^4}
\][/tex]
Now, let's simplify this further by canceling out common terms in the numerator and the denominator.
The term [tex]\(a\)[/tex] in the numerator and [tex]\(a^2\)[/tex] in the denominator will simplify to:
[tex]\[
\frac{a}{a^2} = \frac{1}{a}
\][/tex]
Similarly, the term [tex]\(b\)[/tex] in the numerator and [tex]\(b^4\)[/tex] in the denominator will simplify to:
[tex]\[
\frac{b}{b^4} = \frac{1}{b^3}
\][/tex]
Substituting these simplified terms back into the expression, we get:
[tex]\[
\frac{6ab}{a^2 b^4} = 6 \cdot \frac{1}{a} \cdot \frac{1}{b^3} = \frac{6}{a b^3}
\][/tex]
Thus, the simplified expression equivalent to the given expression is:
[tex]\[
\frac{6}{a b^3}
\][/tex]
Comparing this to the provided options, we find that:
[tex]\[
\boxed{\frac{6}{a b^3}}
\][/tex]
None of the given multiple-choice options (A, B, C, D) match up with this correct simplification directly. Therefore, the correct answer is:
[tex]\[
\boxed{None of the above}
\][/tex]
