Simplify (27a^-9)^(-2/3)

${(27 a^{- 9})}^{- \frac{2}{3}}$

Step 1

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

${(27 \frac{1}{a^{9}})}^{- \frac{2}{3}}$

Step 2

Combine $27$ and $\frac{1}{a^{9}}$ .

${(\frac{27}{a^{9}})}^{- \frac{2}{3}}$

Step 3

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

${(\frac{a^{9}}{27})}^{\frac{2}{3}}$

Step 4

Apply basic rules of exponents.

Apply the product rule to $\frac{a^{9}}{27}$ .

$\frac{{(a^{9})}^{\frac{2}{3}}}{27^{\frac{2}{3}}}$

Multiply the exponents in ${(a^{9})}^{\frac{2}{3}}$ .

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

$\frac{a^{9 (\frac{2}{3})}}{27^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $2$ .

$\frac{a^{6}}{27^{\frac{2}{3}}}$

Step 5

Simplify the denominator.

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

$\frac{a^{6}}{{(3^{3})}^{\frac{2}{3}}}$

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

$\frac{a^{6}}{3^{3 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{a^{6}}{3^{3 (\frac{2}{3})}}$

Rewrite the expression.

$\frac{a^{6}}{3^{2}}$

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

$\frac{a^{6}}{9}$