To evaluate the expression [tex]\(\left( \frac{2^2 x^2 y}{x y^3} \right)^2\)[/tex] given [tex]\(x = 4\)[/tex] and [tex]\(y = 2\)[/tex], we can follow these steps:
1. Simplify the Expression Inside the Parentheses:
Begin with the expression inside the parentheses:
[tex]\[
\frac{2^2 x^2 y}{x y^3}
\][/tex]
2. Substitute the Given Values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 2\)[/tex]:
[tex]\[
\frac{2^2 (4)^2 (2)}{(4) (2)^3}
\][/tex]
3. Compute the Powers and Products:
- 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]
Substituting these values back into the expression:
[tex]\[
\frac{4 \cdot 16 \cdot 2}{4 \cdot 8}
\][/tex]
4. Simplify the Fractions:
Simplify the numerator and the denominator:
[tex]\[
\frac{4 \cdot 16 \cdot 2}{4 \cdot 8} = \frac{128}{32} = 4
\][/tex]
5. Square the Simplified Expression:
Finally, square the simplified result:
[tex]\[
(4)^2 = 16
\][/tex]
Thus, the final result of evaluating the given expression for [tex]\(x = 4\)[/tex] and [tex]\(y = 2\)[/tex] is:
[tex]\[
\boxed{16.0}
\][/tex]
