Sure, let's simplify the given expression step by step:
Given expression:
[tex]\[
(a^b)^{-2} \cdot \left(a^{\frac{1}{3}} \cdot b^2\right)^2 / \left(a^{\frac{1}{6}} \cdot b^{-\frac{1}{2}}\right)
\][/tex]
### Step 1: Simplify each term individually
1. Simplify [tex]\((a^b)^{-2}\)[/tex]:
[tex]\[
(a^b)^{-2} = a^{-2b}
\][/tex]
2. Simplify [tex]\(\left(a^{\frac{1}{3}} \cdot b^2\right)^2\)[/tex]:
[tex]\[
\left(a^{\frac{1}{3}} \cdot b^2\right)^2 = a^{\frac{2}{3}} \cdot b^4
\][/tex]
3. The term [tex]\(\left(a^{\frac{1}{6}} \cdot b^{-\frac{1}{2}}\right)\)[/tex] remains as it is.
So the expression now becomes:
[tex]\[
\frac{a^{-2b} \cdot a^{\frac{2}{3}} \cdot b^4}{a^{\frac{1}{6}} \cdot b^{-\frac{1}{2}}}
\][/tex]
### Step 2: Combine like terms with the same base
To simplify further, we need to combine the terms with the same base (i.e., the terms with base [tex]\(a\)[/tex] and the terms with base [tex]\(b\)[/tex]).
#### For base [tex]\(a\)[/tex]:
Combine the exponents of [tex]\(a\)[/tex]:
[tex]\[
a^{-2b} \cdot a^{\frac{2}{3}} / a^{\frac{1}{6}} = a^{-2b + \frac{2}{3} - \frac{1}{6}}
\][/tex]
Simplify the exponents by converting the fractions to have a common denominator (6):
[tex]\[
-2b + \frac{2}{3} - \frac{1}{6} = -2b + \frac{4}{6} - \frac{1}{6} = -2b + \frac{3}{6} = -2b + \frac{1}{2}
\][/tex]
So the exponent of [tex]\(a\)[/tex] becomes:
[tex]\[
-2b + \frac{1}{2}
\][/tex]
#### For base [tex]\(b\)[/tex]:
Combine the exponents of [tex]\(b\)[/tex]:
[tex]\[
b^4 / b^{-\frac{1}{2}} = b^{4 - \left(-\frac{1}{2}\right)} = b^{4 + \frac{1}{2}} = b^{4.5}
\][/tex]
### Step 3: Combine the simplified terms
So, putting it all together, we have:
[tex]\[
a^{-2b + \frac{1}{2}} \cdot b^{4.5}
\][/tex]
Therefore, the fully simplified expression is:
[tex]\[
a^{-1.5} \cdot b^{4.5}
\][/tex]
So the exponents for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
\begin{align}
a: & \quad -1.5 \\
b: & \quad 4.5
\end{align}
