Simplify (27a^-3)^(-2/3)

${(27 a^{- 3})}^{- \frac{2}{3}}$

Step 1

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

${(27 \frac{1}{a^{3}})}^{- \frac{2}{3}}$

Step 2

Combine $27$ and $\frac{1}{a^{3}}$ .

${(\frac{27}{a^{3}})}^{- \frac{2}{3}}$

Step 3

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

${(\frac{a^{3}}{27})}^{\frac{2}{3}}$

Step 4

Apply basic rules of exponents.

Apply the product rule to $\frac{a^{3}}{27}$ .

$\frac{{(a^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}}}$

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

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

$\frac{a^{3 (\frac{2}{3})}}{27^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{a^{3 (\frac{2}{3})}}{27^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{a^{2}}{27^{\frac{2}{3}}}$

Step 5

Simplify the denominator.

Rewrite $27$ as $3^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$\frac{a^{2}}{9}$