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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

Apply the product rule to $3 y^{4}$ .

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

Step 6

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

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

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

Multiply $2$ by $3$ .

$\frac{x^{6}}{3^{3} {(y^{4})}^{3}}$

Step 7

Simplify the denominator.

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

$\frac{x^{6}}{27 {(y^{4})}^{3}}$

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

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

$\frac{x^{6}}{27 y^{4 \cdot 3}}$

Multiply $4$ by $3$ .

$\frac{x^{6}}{27 y^{12}}$