Simplify (2x^-3y^4)^-2

${(2 x^{- 3} y^{4})}^{- 2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(2 \frac{1}{x^{3}} y^{4})}^{- 2}$

Step 2

Combine $2$ and $\frac{1}{x^{3}}$ .

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

Step 3

Combine $\frac{2}{x^{3}}$ and $y^{4}$ .

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

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 5

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

Apply the product rule to $\frac{x^{3}}{2 y^{4}}$ .

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

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

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

Step 6

Multiply the exponents in ${(x^{3})}^{2}$ .

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

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

Multiply $3$ by $2$ .

$\frac{x^{6}}{2^{2} {(y^{4})}^{2}}$

Step 7

Simplify the denominator.

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

$\frac{x^{6}}{4 {(y^{4})}^{2}}$

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

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

$\frac{x^{6}}{4 y^{4 \cdot 2}}$

Multiply $4$ by $2$ .

$\frac{x^{6}}{4 y^{8}}$