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

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

Step 1

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

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

Step 2

Combine $4$ and $\frac{1}{x^{4}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

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

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

Step 6

Multiply the exponents in ${(x^{4})}^{3}$ .

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

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

Multiply $4$ by $3$ .

$\frac{x^{12}}{4^{3} {(y^{2})}^{3}}$

Step 7

Simplify the denominator.

Raise $4$ to the power of $3$ .

$\frac{x^{12}}{64 {(y^{2})}^{3}}$

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

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

$\frac{x^{12}}{64 y^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{x^{12}}{64 y^{6}}$