Simplify cube root of 24n^2* cube root of 36n^2

$\sqrt[3]{24 n^{2}} \cdot \sqrt[3]{36 n^{2}}$

Step 1

Rewrite $24 n^{2}$ as $2^{3} \cdot (3 n^{2})$ .

Factor $8$ out of $24$ .

$\sqrt[3]{8 (3) n^{2}} \cdot \sqrt[3]{36 n^{2}}$

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3} \cdot 3 n^{2}} \cdot \sqrt[3]{36 n^{2}}$

Add parentheses.

$\sqrt[3]{2^{3} \cdot (3 n^{2})} \cdot \sqrt[3]{36 n^{2}}$

Step 2

Pull terms out from under the radical.

$2 \sqrt[3]{3 n^{2}} \cdot \sqrt[3]{36 n^{2}}$

Step 3

Multiply $2 \sqrt[3]{3 n^{2}} \sqrt[3]{36 n^{2}}$ .

Combine using the product rule for radicals.

$2 \sqrt[3]{36 n^{2} (3 n^{2})}$

Multiply $3$ by $36$ .

$2 \sqrt[3]{108 n^{2} n^{2}}$

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

Move $n^{2}$ .

$2 \sqrt[3]{108 (n^{2} n^{2})}$

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

$2 \sqrt[3]{108 n^{2 + 2}}$

Add $2$ and $2$ .

$2 \sqrt[3]{108 n^{4}}$

Step 4

Rewrite $108 n^{4}$ as ${(3 n)}^{3} \cdot (4 n)$ .

Factor $27$ out of $108$ .

$2 \sqrt[3]{27 (4) n^{4}}$

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

$2 \sqrt[3]{3^{3} \cdot 4 n^{4}}$

Factor out $n^{3}$ .

$2 \sqrt[3]{3^{3} \cdot 4 (n^{3} n)}$

Move $4$ .

$2 \sqrt[3]{3^{3} n^{3} \cdot 4 n}$

Rewrite $3^{3} n^{3}$ as ${(3 n)}^{3}$ .

$2 \sqrt[3]{{(3 n)}^{3} \cdot 4 n}$

Add parentheses.

$2 \sqrt[3]{{(3 n)}^{3} \cdot (4 n)}$

Step 5

Pull terms out from under the radical.

$2 (3 n \sqrt[3]{4 n})$

Step 6

Multiply $3$ by $2$ .

$6 n \sqrt[3]{4 n}$