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

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

Add $2$ and $4$ .

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

Step 4

Simplify the expression.

Raise $2$ to the power of $6$ .

${(\frac{64}{4} \cdot {(2)}^{3})}^{2}$

Divide $64$ by $4$ .

${(16 \cdot {(2)}^{3})}^{2}$

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

${(16 \cdot 8)}^{2}$

Multiply $16$ by $8$ .

$128^{2}$

Raise $128$ to the power of $2$ .

$16384$