Simplify cube root of x^13 ninth root of x^6

$\sqrt[3]{x^{13}} \sqrt[9]{x^{6}}$

Step 1

Rewrite $x^{13}$ as ${(x^{4})}^{3} x$ .

Factor out $x^{12}$ .

$\sqrt[3]{x^{12} x} \sqrt[9]{x^{6}}$

Rewrite $x^{12}$ as ${(x^{4})}^{3}$ .

$\sqrt[3]{{(x^{4})}^{3} x} \sqrt[9]{x^{6}}$

Step 2

Pull terms out from under the radical.

$x^{4} \sqrt[3]{x} \sqrt[9]{x^{6}}$

Step 3

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

$x^{4} \sqrt[3]{x} \sqrt[9]{{(x^{2})}^{3}}$

Step 4

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

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

Step 5

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

$x^{4} \sqrt[3]{x} \sqrt[3]{x^{2}}$

Step 6

Multiply $x^{4} \sqrt[3]{x} \sqrt[3]{x^{2}}$ .

Combine using the product rule for radicals.

$x^{4} \sqrt[3]{x^{2} x}$

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

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

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

$x^{4} \sqrt[3]{x^{2} x^{1}}$

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

$x^{4} \sqrt[3]{x^{2 + 1}}$

Add $2$ and $1$ .

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

Step 7

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

$x^{4} x$

Step 8

Multiply $x^{4}$ by $x$ by adding the exponents.

Multiply $x^{4}$ by $x$ .

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

$x^{4} x^{1}$

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

$x^{4 + 1}$

Add $4$ and $1$ .

$x^{5}$