Simplify (8a^-3)^(-2/3)

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

Step 1

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

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

Step 2

Combine $8$ and $\frac{1}{a^{3}}$ .

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

Step 3

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

${(\frac{a^{3}}{8})}^{\frac{2}{3}}$

Step 4

Apply basic rules of exponents.

Apply the product rule to $\frac{a^{3}}{8}$ .

$\frac{{(a^{3})}^{\frac{2}{3}}}{8^{\frac{2}{3}}}$

Multiply the exponents in ${(a^{3})}^{\frac{2}{3}}$ .

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

$\frac{a^{3 (\frac{2}{3})}}{8^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{a^{3 (\frac{2}{3})}}{8^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{a^{2}}{8^{\frac{2}{3}}}$

Step 5

Simplify the denominator.

Rewrite $8$ as $2^{3}$ .

$\frac{a^{2}}{{(2^{3})}^{\frac{2}{3}}}$

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

$\frac{a^{2}}{2^{3 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{a^{2}}{2^{3 (\frac{2}{3})}}$

Rewrite the expression.

$\frac{a^{2}}{2^{2}}$

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

$\frac{a^{2}}{4}$