Let's simplify the given expression step-by-step:
The expression is:
[tex]\[
\left(\frac{\left(2^{-3}\right)\left(x^{-3}\right)\left(y^2\right)}{\left(4^{-2}\right)\left(x^4\right)\left(y^6\right)}\right)^2
\][/tex]
First, let's simplify the fraction inside the parentheses.
### Numerator:
[tex]\[
2^{-3} \cdot x^{-3} \cdot y^2
\][/tex]
### Denominator:
[tex]\[
4^{-2} \cdot x^4 \cdot y^6
\][/tex]
#### Simplify [tex]\( 4^{-2} \)[/tex]:
Note that [tex]\( 4 = 2^2 \)[/tex], so:
[tex]\[
4^{-2} = (2^2)^{-2} = 2^{-4}
\][/tex]
Therefore, the denominator becomes:
[tex]\[
2^{-4} \cdot x^4 \cdot y^6
\][/tex]
Now we rewrite the entire fraction:
[tex]\[
\frac{2^{-3} \cdot x^{-3} \cdot y^2}{2^{-4} \cdot x^4 \cdot y^6}
\][/tex]
### Simplify the fraction by dividing the corresponding bases:
#### For the base 2:
[tex]\[
\frac{2^{-3}}{2^{-4}} = 2^{-3 - (-4)} = 2^{-3 + 4} = 2^1 = 2
\][/tex]
#### For the base x:
[tex]\[
\frac{x^{-3}}{x^4} = x^{-3 - 4} = x^{-7}
\][/tex]
#### For the base y:
[tex]\[
\frac{y^2}{y^6} = y^{2 - 6} = y^{-4}
\][/tex]
So, the fraction simplifies to:
[tex]\[
2 \cdot x^{-7} \cdot y^{-4}
\][/tex]
Now, let's square the entire simplified expression:
[tex]\[
\left(2 \cdot x^{-7} \cdot y^{-4}\right)^2
\][/tex]
### Squaring each part separately:
#### For the coefficient 2:
[tex]\[
(2)^2 = 4
\][/tex]
#### For the base x:
[tex]\[
(x^{-7})^2 = x^{-7 \cdot 2} = x^{-14}
\][/tex]
#### For the base y:
[tex]\[
(y^{-4})^2 = y^{-4 \cdot 2} = y^{-8}
\][/tex]
Combining these results, we get:
[tex]\[
4 \cdot x^{-14} \cdot y^{-8} = \frac{4}{x^{14} \cdot y^8}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{4}{x^{14} y^8}}
\][/tex]
