Simplify (2x^3y^-8)^-3

${(2 x^{3} y^{- 8})}^{- 3}$

Step 1

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

${(2 x^{3} \frac{1}{y^{8}})}^{- 3}$

Step 2

Multiply $2 x^{3} \frac{1}{y^{8}}$ .

Combine $2$ and $\frac{1}{y^{8}}$ .

${(x^{3} \frac{2}{y^{8}})}^{- 3}$

Combine $x^{3}$ and $\frac{2}{y^{8}}$ .

${(\frac{x^{3} \cdot 2}{y^{8}})}^{- 3}$

Step 3

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

${(\frac{2 \cdot x^{3}}{y^{8}})}^{- 3}$

Step 4

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

${(\frac{y^{8}}{2 x^{3}})}^{3}$

Step 5

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

Apply the product rule to $\frac{y^{8}}{2 x^{3}}$ .

$\frac{{(y^{8})}^{3}}{{(2 x^{3})}^{3}}$

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

$\frac{{(y^{8})}^{3}}{2^{3} {(x^{3})}^{3}}$

Step 6

Multiply the exponents in ${(y^{8})}^{3}$ .

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

$\frac{y^{8 \cdot 3}}{2^{3} {(x^{3})}^{3}}$

Multiply $8$ by $3$ .

$\frac{y^{24}}{2^{3} {(x^{3})}^{3}}$

Step 7

Simplify the denominator.

Raise $2$ to the power of $3$ .

$\frac{y^{24}}{8 {(x^{3})}^{3}}$

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

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

$\frac{y^{24}}{8 x^{3 \cdot 3}}$

Multiply $3$ by $3$ .

$\frac{y^{24}}{8 x^{9}}$