Simplify ( cube root of 27a^9)^2

${(\sqrt[3]{27 a^{9}})}^{2}$

Rewrite $27 a^{9}$ as ${(3 a^{3})}^{3}$ .

${\sqrt[3]{{(3 a^{3})}^{3}}}^{2}$

Pull terms out from under the radical, assuming real numbers.

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

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

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

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

$9 {(a^{3})}^{2}$

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

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

$9 a^{3 \cdot 2}$

Multiply $3$ by $2$ .

$9 a^{6}$