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

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

Step 1

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

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

Step 2

Multiply $2 x^{3} \frac{1}{y^{3}}$ .

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

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

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

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

Step 3

Move $2$ to the left of $x^{3}$ .

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

Step 4

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

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

Step 5

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

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

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

Apply the product rule to $2 x^{3}$ .

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

Step 6

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

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

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

Multiply $3$ by $2$ .

$\frac{y^{6}}{2^{2} {(x^{3})}^{2}}$

Step 7

Simplify the denominator.

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

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

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

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

$\frac{y^{6}}{4 x^{3 \cdot 2}}$

Multiply $3$ by $2$ .

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