Evaluate Using the Given Value ((2^2x^2y)/(xy^3))^2 for x=4 and y=2

${(\frac{2^{2} x^{2} y}{x y^{3}})}^{2}$ for $x = 4$ and $y = 2$

Replace the variable $x$ with $4$ in the expression.

${(\frac{2^{2} {(4)}^{2} y}{4} \cdot y^{3})}^{2}$

Replace the variable $y$ with $2$ in the expression.

${(\frac{2^{2} {(4)}^{2} (2)}{4} \cdot {(2)}^{3})}^{2}$

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

${(\frac{2^{2} \cdot {(2^{2})}^{2} \cdot 2}{4} \cdot {(2)}^{3})}^{2}$

Multiply the exponents in ${(2^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{2^{2} \cdot 2^{2 \cdot 2} \cdot 2}{4} \cdot {(2)}^{3})}^{2}$

Multiply $2$ by $2$ .

${(\frac{2^{2} \cdot 2^{4} \cdot 2}{4} \cdot {(2)}^{3})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2^{2 + 4} \cdot 2}{4} \cdot {(2)}^{3})}^{2}$

Add $2$ and $4$ .

${(\frac{2^{6} \cdot 2}{4} \cdot {(2)}^{3})}^{2}$

Multiply $2^{6}$ by $2$ by adding the exponents.

Multiply $2^{6}$ by $2$ .

Raise $2$ to the power of $1$ .

${(\frac{2^{6} \cdot 2^{1}}{4} \cdot {(2)}^{3})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2^{6 + 1}}{4} \cdot {(2)}^{3})}^{2}$

Add $6$ and $1$ .

${(\frac{2^{7}}{4} \cdot {(2)}^{3})}^{2}$

Simplify the expression.

Raise $2$ to the power of $7$ .

${(\frac{128}{4} \cdot {(2)}^{3})}^{2}$

Divide $128$ by $4$ .

${(32 \cdot {(2)}^{3})}^{2}$

Raise $2$ to the power of $3$ .

${(32 \cdot 8)}^{2}$

Multiply $32$ by $8$ .

$256^{2}$

Raise $256$ to the power of $2$ .

$65536$