Simplify cube root of 9x^4* cube root of 3x^8

$\sqrt[3]{9 x^{4}} \cdot \sqrt[3]{3 x^{8}}$

Step 1

Rewrite $9 x^{4}$ as $x^{3} \cdot (9 x)$ .

Factor out $x^{3}$ .

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

Reorder $9$ and $x^{3}$ .

$\sqrt[3]{x^{3} \cdot 9 x} \cdot \sqrt[3]{3 x^{8}}$

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$x \sqrt[3]{9 x} \cdot \sqrt[3]{3 x^{8}}$

Step 3

Rewrite $3 x^{8}$ as ${(x^{2})}^{3} \cdot (3 x^{2})$ .

Factor out $x^{6}$ .

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

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

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

Reorder $3$ and ${(x^{2})}^{3}$ .

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Move $x^{2}$ .

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

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

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

$x^{2} x^{1} \sqrt[3]{9 x} \cdot \sqrt[3]{3 x^{2}}$

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

$x^{2 + 1} \sqrt[3]{9 x} \cdot \sqrt[3]{3 x^{2}}$

Add $2$ and $1$ .

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

Step 6

Multiply $x^{3} \sqrt[3]{9 x} \sqrt[3]{3 x^{2}}$ .

Combine using the product rule for radicals.

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

Multiply $9$ by $3$ .

$x^{3} \sqrt[3]{27 x^{2} x}$

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

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

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

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

Add $1$ and $2$ .

$x^{3} \sqrt[3]{27 x^{3}}$

Step 7

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

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

Step 8

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

$x^{3} (3 x)$

Step 9

Rewrite using the commutative property of multiplication.

$3 x^{3} x$

Step 10

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

Move $x$ .

$3 (x \cdot x^{3})$

Multiply $x$ by $x^{3}$ .

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

$3 (x^{1} x^{3})$

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

$3 x^{1 + 3}$

Add $1$ and $3$ .

$3 x^{4}$

Simplify cube root of 9x^4* cube root of 3x^8

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