Algebra Examples

$\frac{\sqrt[3]{y^{4}}}{\sqrt[6]{y^{4}}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out $y^{3}$ .

$\frac{\sqrt[3]{y^{3} y}}{\sqrt[6]{y^{4}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{y \sqrt[3]{y}}{\sqrt[6]{y^{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{y \sqrt[3]{y}}{\sqrt[6]{{(y^{2})}^{2}}}$

Step 2.2

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

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

Step 2.3

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

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

Step 3

Combine $\sqrt[3]{y}$ and $\sqrt[3]{y^{2}}$ into a single radical.

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

Step 4

Reduce the expression $\frac{y}{y^{2}}$ by cancelling the common factors.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$y \sqrt[3]{\frac{y \cdot 1}{y \cdot y}}$

Step 4.4

Cancel the common factor.

$y \sqrt[3]{\frac{y \cdot 1}{y \cdot y}}$

Step 4.5

Rewrite the expression.

$y \sqrt[3]{\frac{1}{y}}$

Step 5

Rewrite $\sqrt[3]{\frac{1}{y}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{y}}$ .

$y \frac{\sqrt[3]{1}}{\sqrt[3]{y}}$

Step 6

Any root of $1$ is $1$ .

$y \frac{1}{\sqrt[3]{y}}$

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

Raise $\sqrt[3]{y}$ to the power of $1$ .

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

Step 8.3

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

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

Step 8.4

Add $1$ and $2$ .

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

Step 8.5

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

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Step 8.5.1

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

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

Step 8.5.2

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

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

Step 8.5.3

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

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

Step 8.5.4

Cancel the common factor of $3$ .

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Step 8.5.4.1

Cancel the common factor.

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

Step 8.5.4.2

Rewrite the expression.

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

Step 8.5.5

Simplify.

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

Step 9

Cancel the common factor of $y$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

${\sqrt[3]{y}}^{2}$

Step 10

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

$\sqrt[3]{y^{2}}$