Evaluate ((2x^0)/(4x^-2y^4))^2

${(\frac{2 x^{0}}{4 x^{- 2} y^{4}})}^{2}$

Step 1

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 x^{0} x^{2}}{4 y^{4}})}^{2}$

Step 2

Multiply $x^{0}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

${(\frac{2 (x^{2} x^{0})}{4 y^{4}})}^{2}$

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

${(\frac{2 x^{2 + 0}}{4 y^{4}})}^{2}$

Add $2$ and $0$ .

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

Step 3

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 x^{2}$ .

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

Cancel the common factors.

Factor $2$ out of $4 y^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{x^{2}}{2 y^{4}}$ .

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

Apply the product rule to $2 y^{4}$ .

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

Step 5

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

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

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

Multiply $2$ by $2$ .

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

Step 6

Simplify the denominator.

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

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

Multiply the exponents in ${(y^{4})}^{2}$ .

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

$\frac{x^{4}}{4 y^{4 \cdot 2}}$

Multiply $4$ by $2$ .

$\frac{x^{4}}{4 y^{8}}$