Find the Product cube root of 16x^7* cube root of 12x^9

$\sqrt[3]{16 x^{7}} \cdot \sqrt[3]{12 x^{9}}$

Step 1

Rewrite $16 x^{7}$ as ${(2 x^{2})}^{3} \cdot (2 x)$ .

Factor $8$ out of $16$ .

$\sqrt[3]{8 (2) x^{7}} \cdot \sqrt[3]{12 x^{9}}$

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

$\sqrt[3]{2^{3} \cdot 2 x^{7}} \cdot \sqrt[3]{12 x^{9}}$

Factor out $x^{6}$ .

$\sqrt[3]{2^{3} \cdot 2 (x^{6} x)} \cdot \sqrt[3]{12 x^{9}}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\sqrt[3]{2^{3} \cdot 2 ({(x^{2})}^{3} x)} \cdot \sqrt[3]{12 x^{9}}$

Move $2$ .

$\sqrt[3]{2^{3} {(x^{2})}^{3} \cdot 2 x} \cdot \sqrt[3]{12 x^{9}}$

Rewrite $2^{3} {(x^{2})}^{3}$ as ${(2 x^{2})}^{3}$ .

$\sqrt[3]{{(2 x^{2})}^{3} \cdot 2 x} \cdot \sqrt[3]{12 x^{9}}$

Add parentheses.

$\sqrt[3]{{(2 x^{2})}^{3} \cdot (2 x)} \cdot \sqrt[3]{12 x^{9}}$

Step 2

Pull terms out from under the radical.

$2 x^{2} \sqrt[3]{2 x} \cdot \sqrt[3]{12 x^{9}}$

Step 3

Rewrite $12 x^{9}$ as ${(x^{3})}^{3} \cdot 12$ .

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

$2 x^{2} \sqrt[3]{2 x} \cdot \sqrt[3]{12 {(x^{3})}^{3}}$

Reorder $12$ and ${(x^{3})}^{3}$ .

$2 x^{2} \sqrt[3]{2 x} \cdot \sqrt[3]{{(x^{3})}^{3} \cdot 12}$

Step 4

Pull terms out from under the radical.

$2 x^{2} \sqrt[3]{2 x} \cdot (x^{3} \sqrt[3]{12})$

Step 5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$2 (x^{3} x^{2}) \sqrt[3]{2 x} \cdot \sqrt[3]{12}$

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

$2 x^{3 + 2} \sqrt[3]{2 x} \cdot \sqrt[3]{12}$

Add $3$ and $2$ .

$2 x^{5} \sqrt[3]{2 x} \cdot \sqrt[3]{12}$

Step 6

Multiply $2 x^{5} \sqrt[3]{2 x} \sqrt[3]{12}$ .

Combine using the product rule for radicals.

$2 x^{5} \sqrt[3]{12 (2 x)}$

Multiply $12$ by $2$ .

$2 x^{5} \sqrt[3]{24 x}$

Step 7

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

Factor $8$ out of $24$ .

$2 x^{5} \sqrt[3]{8 (3) x}$

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

$2 x^{5} \sqrt[3]{2^{3} \cdot 3 x}$

Add parentheses.

$2 x^{5} \sqrt[3]{2^{3} \cdot (3 x)}$

Step 8

Pull terms out from under the radical.

$2 x^{5} (2 \sqrt[3]{3 x})$

Step 9

Multiply $2$ by $2$ .

$4 x^{5} \sqrt[3]{3 x}$

Find the Product cube root of 16x^7* cube root of 12x^9

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