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

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

Move $b^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

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

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

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

Apply the product rule to $\frac{3}{a^{3} b^{2}}$ .

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

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

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

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

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

Simplify the denominator.

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

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

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

Multiply $3$ by $3$ .

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

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

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

$\frac{27}{a^{9} b^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{27}{a^{9} b^{6}}$