Algebra Examples

$\frac{\sqrt[6]{4 x}}{\sqrt[6]{16 x^{5}}}$

Step 1

Combine $\sqrt[6]{4 x}$ and $\sqrt[6]{16 x^{5}}$ into a single radical.

$\sqrt[6]{\frac{4 x}{16 x^{5}}}$

Step 2

Reduce the expression $\frac{4 x}{16 x^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $4$ out of $4 x$ .

$\sqrt[6]{\frac{4 (x)}{16 x^{5}}}$

Step 2.2

Factor $4$ out of $16 x^{5}$ .

$\sqrt[6]{\frac{4 (x)}{4 (4 x^{5})}}$

Step 2.3

Cancel the common factor.

$\sqrt[6]{\frac{4 x}{4 (4 x^{5})}}$

Step 2.4

Rewrite the expression.

$\sqrt[6]{\frac{x}{4 x^{5}}}$

Step 3

Cancel the common factor of $x$ and $x^{5}$ .

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Step 3.1

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

$\sqrt[6]{\frac{x^{1}}{4 x^{5}}}$

Step 3.2

Factor $x$ out of $x^{1}$ .

$\sqrt[6]{\frac{x \cdot 1}{4 x^{5}}}$

Step 3.3

Cancel the common factors.

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Step 3.3.1

Factor $x$ out of $4 x^{5}$ .

$\sqrt[6]{\frac{x \cdot 1}{x (4 x^{4})}}$

Step 3.3.2

Cancel the common factor.

$\sqrt[6]{\frac{x \cdot 1}{x (4 x^{4})}}$

Step 3.3.3

Rewrite the expression.

$\sqrt[6]{\frac{1}{4 x^{4}}}$

Step 4

Rewrite $1$ as $1^{2}$ .

$\sqrt[6]{\frac{1^{2}}{4 x^{4}}}$

Step 5

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

$\sqrt[6]{\frac{1^{2}}{{(2 x^{2})}^{2}}}$

Step 6

Rewrite $\frac{1^{2}}{{(2 x^{2})}^{2}}$ as ${(\frac{1}{2 x^{2}})}^{2}$ .

$\sqrt[6]{{(\frac{1}{2 x^{2}})}^{2}}$

Step 7

Rewrite $\sqrt[6]{{(\frac{1}{2 x^{2}})}^{2}}$ as $\sqrt[3]{\sqrt{{(\frac{1}{2 x^{2}})}^{2}}}$ .

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

Step 8

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

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

Step 9

Rewrite $\sqrt[3]{\frac{1}{2 x^{2}}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{2 x^{2}}}$ .

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

Step 10

Any root of $1$ is $1$ .

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

Step 11

Multiply $\frac{1}{\sqrt[3]{2 x^{2}}}$ by $\frac{{\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{2}}$ .

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

Step 12

Combine and simplify the denominator.

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Step 12.1

Multiply $\frac{1}{\sqrt[3]{2 x^{2}}}$ by $\frac{{\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{2}}$ .

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

Step 12.2

Raise $\sqrt[3]{2 x^{2}}$ to the power of $1$ .

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

Step 12.3

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

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

Step 12.4

Add $1$ and $2$ .

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

Step 12.5

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

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Step 12.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x^{2}}$ as ${(2 x^{2})}^{\frac{1}{3}}$ .

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

Step 12.5.2

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

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

Step 12.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 12.5.4

Cancel the common factor of $3$ .

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Step 12.5.4.1

Cancel the common factor.

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

Step 12.5.4.2

Rewrite the expression.

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

Step 12.5.5

Simplify.

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

Step 13

Simplify the numerator.

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Step 13.1

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

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

Step 13.2

Apply the product rule to $2 x^{2}$ .

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

Step 13.3

Raise $2$ to the power of $2$ .

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

Step 13.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 13.4.1

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

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

Step 13.4.2

Multiply $2$ by $2$ .

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

Step 13.5

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

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Step 13.5.1

Factor out $x^{3}$ .

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

Step 13.5.2

Reorder $4$ and $x^{3}$ .

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

Step 13.5.3

Add parentheses.

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

Step 13.6

Pull terms out from under the radical.

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

Step 14

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

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Step 14.1

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

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

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $x$ out of $2 x^{2}$ .

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

Step 14.2.2

Cancel the common factor.

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

Step 14.2.3

Rewrite the expression.

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