To evaluate the expression [tex]\(\left(\frac{2^2 x^2}{x y^3}\right)^2\)[/tex] for [tex]\(x = 4\)[/tex] and [tex]\(y = 2\)[/tex], we will follow these steps:
1. Substitute the given values into the expression:
[tex]\[
\left(\frac{2^2 \cdot 4^2}{4 \cdot 2^3}\right)^2
\][/tex]
2. Simplify the inner expression:
- Calculate [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
- Calculate [tex]\(4^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
- Calculate [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
Now substitute the calculated values back into the expression:
[tex]\[
\left(\frac{4 \cdot 16}{4 \cdot 8}\right)^2
\][/tex]
3. Simplify the fraction inside the parentheses:
- Multiply the numbers in the numerator:
[tex]\[
4 \cdot 16 = 64
\][/tex]
- Multiply the numbers in the denominator:
[tex]\[
4 \cdot 8 = 32
\][/tex]
Now we have:
[tex]\[
\left(\frac{64}{32}\right)^2
\][/tex]
4. Simplify the fraction [tex]\(\frac{64}{32}\)[/tex]:
[tex]\[
\frac{64}{32} = 2
\][/tex]
5. Square the result:
[tex]\[
2^2 = 4
\][/tex]
6. Conclusion:
The evaluated expression [tex]\(\left(\frac{2^2 x^2}{x y^3}\right)^2\)[/tex] for [tex]\(x = 4\)[/tex] and [tex]\(y = 2\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
