Simplify cube root of 9a* cube root of 3a^2b^3

$\sqrt[3]{9 a} \cdot \sqrt[3]{3 a^{2} b^{3}}$

Rewrite $3 a^{2} b^{3}$ as $b^{3} \cdot (3 a^{2})$ .

Move $a^{2}$ .

$\sqrt[3]{9 a} \cdot \sqrt[3]{3 b^{3} a^{2}}$

Reorder $3$ and $b^{3}$ .

$\sqrt[3]{9 a} \cdot \sqrt[3]{b^{3} \cdot 3 a^{2}}$

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $\sqrt[3]{9 a} (b \sqrt[3]{3 a^{2}})$ .

Combine using the product rule for radicals.

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

Multiply $3$ by $9$ .

$b \sqrt[3]{27 a \cdot a^{2}}$

Multiply $a$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

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

Multiply $a^{2}$ by $a$ .

Raise $a$ to the power of $1$ .

$b \sqrt[3]{27 (a^{2} a^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b \sqrt[3]{27 a^{2 + 1}}$

Add $2$ and $1$ .

$b \sqrt[3]{27 a^{3}}$

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

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

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

$b (3 a)$

Rewrite using the commutative property of multiplication.

$3 b a$