Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Any root of $1$ is $1$ .

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

Step 6

Multiply $2$ by $1$ .

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $3$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $4$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

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

Step 9

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

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Step 9.1

Move $x^{3}$ .

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

Add $3$ and $1$ .

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

Step 10

Simplify the numerator.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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Step 10.4.1

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

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

Step 10.4.2

Multiply $3$ by $3$ .

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

Step 10.5

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

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Step 10.5.1

Factor $81$ out of $729$ .

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

Step 10.5.2

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

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

Step 10.5.3

Factor out $x^{8}$ .

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

Step 10.5.4

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

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

Step 10.5.5

Move $9$ .

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

Step 10.5.6

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

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

Step 10.5.7

Add parentheses.

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

Step 10.6

Pull terms out from under the radical.

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

Step 10.7

Multiply $2$ by $3$ .

$\frac{6 x^{2} \sqrt[4]{9 x}}{x^{4} \cdot 9}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

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

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Step 11.1.1

Factor $3$ out of $6 x^{2} \sqrt[4]{9 x}$ .

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

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

Factor $3$ out of $x^{4} \cdot 9$ .

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

Step 11.1.2.2

Cancel the common factor.

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

Step 11.1.2.3

Rewrite the expression.

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

Step 11.2

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 11.2.1

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

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

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $x^{2}$ out of $x^{4} \cdot 3$ .

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

Step 11.2.2.2

Cancel the common factor.

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

Step 11.2.2.3

Rewrite the expression.

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

Step 11.3

Move $3$ to the left of $x^{2}$ .

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