Simplify fourth root of 16/(9x^7)

$\sqrt[4]{\frac{16}{9 x^{7}}}$

Rewrite $\frac{16}{9 x^{7}}$ as ${(\frac{2}{x})}^{4} \frac{1}{9 x^{3}}$ .

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Factor the perfect power $2^{4}$ out of $16$ .

$\sqrt[4]{\frac{2^{4} \cdot 1}{9 x^{7}}}$

Factor the perfect power $x^{4}$ out of $9 x^{7}$ .

$\sqrt[4]{\frac{2^{4} \cdot 1}{x^{4} \cdot (9 x^{3})}}$

Rearrange the fraction $\frac{2^{4} \cdot 1}{x^{4} \cdot (9 x^{3})}$ .

$\sqrt[4]{{(\frac{2}{x})}^{4} \frac{1}{9 x^{3}}}$

Pull terms out from under the radical.

$\frac{2}{x} \sqrt[4]{\frac{1}{9 x^{3}}}$

Rewrite $\sqrt[4]{\frac{1}{9 x^{3}}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{9 x^{3}}}$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{1}}{\sqrt[4]{9 x^{3}}}$

Any root of $1$ is $1$ .

$\frac{2}{x} \cdot \frac{1}{\sqrt[4]{9 x^{3}}}$

Multiply $\frac{1}{\sqrt[4]{9 x^{3}}}$ by $\frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{3}}$ .

$\frac{2}{x} (\frac{1}{\sqrt[4]{9 x^{3}}} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{3}})$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt[4]{9 x^{3}}}$ and $\frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{3}}$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{\sqrt[4]{9 x^{3}} {\sqrt[4]{9 x^{3}}}^{3}}$

Raise $\sqrt[4]{9 x^{3}}$ to the power of $1$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{1} {\sqrt[4]{9 x^{3}}}^{3}}$

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

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{\sqrt[4]{9 x^{3}}}^{4}}$

Rewrite ${\sqrt[4]{9 x^{3}}}^{4}$ as $9 x^{3}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9 x^{3}}$ as ${(9 x^{3})}^{\frac{1}{4}}$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{({(9 x^{3})}^{\frac{1}{4}})}^{4}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{(9 x^{3})}^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{{(9 x^{3})}^{1}}$

Simplify.

$\frac{2}{x} \cdot \frac{{\sqrt[4]{9 x^{3}}}^{3}}{9 x^{3}}$

Simplify the numerator.

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Rewrite ${\sqrt[4]{9 x^{3}}}^{3}$ as ${({(9 x^{3})}^{3})}^{\frac{1}{4}}$ .

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

Apply the product rule to $9 x^{3}$ .

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

Raise $9$ to the power of $3$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{729 {(x^{3})}^{3}}}{9 x^{3}}$

Multiply the exponents in ${(x^{3})}^{3}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{729 x^{3 \cdot 3}}}{9 x^{3}}$

Multiply $3$ by $3$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{729 x^{9}}}{9 x^{3}}$

Rewrite $729 x^{9}$ as ${(3 x^{2})}^{4} \cdot (9 x)$ .

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Factor $81$ out of $729$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{81 (9) x^{9}}}{9 x^{3}}$

Rewrite $81$ as $3^{4}$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{3^{4} \cdot 9 x^{9}}}{9 x^{3}}$

Factor out $x^{8}$ .

$\frac{2}{x} \cdot \frac{\sqrt[4]{3^{4} \cdot 9 (x^{8} x)}}{9 x^{3}}$

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

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

Move $9$ .

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

Rewrite $3^{4} {(x^{2})}^{4}$ as ${(3 x^{2})}^{4}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

$\frac{2}{x} \cdot \frac{3 x^{2} \sqrt[4]{9 x}}{9 x^{3}}$

Simplify terms.

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Cancel the common factor of $x$ .

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Factor $x$ out of $3 x^{2} \sqrt[4]{9 x}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $2$ and $\frac{3 x \sqrt[4]{9 x}}{9 x^{3}}$ .

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

Multiply $3$ by $2$ .

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

Cancel the common factor of $6$ and $9$ .

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Factor $3$ out of $6 (x \sqrt[4]{9 x})$ .

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

Cancel the common factors.

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Factor $3$ out of $9 x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ and $x^{3}$ .

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Factor $x$ out of $2 (x \sqrt[4]{9 x})$ .

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

Cancel the common factors.

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Factor $x$ out of $3 x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Simplify fourth root of 16/(9x^7)

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