Algebra Examples

$\sqrt[20]{\frac{y^{- 8}}{y^{- 3}}}$

Step 1

Reduce the expression $\frac{y^{- 8}}{y^{- 3}}$ by cancelling the common factors.

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Step 1.1

Multiply by $1$ .

$\sqrt[20]{\frac{y^{- 8} \cdot 1}{y^{- 3}}}$

Step 1.2

Factor $y^{- 8}$ out of $y^{- 3}$ .

$\sqrt[20]{\frac{y^{- 8} \cdot 1}{y^{- 8} y^{5}}}$

Step 1.3

Cancel the common factor.

$\sqrt[20]{\frac{y^{- 8} \cdot 1}{y^{- 8} y^{5}}}$

Step 1.4

Rewrite the expression.

$\sqrt[20]{\frac{1}{y^{5}}}$

Step 2

Rewrite $1$ as $1^{5}$ .

$\sqrt[20]{\frac{1^{5}}{y^{5}}}$

Step 3

Rewrite $\frac{1^{5}}{y^{5}}$ as ${(\frac{1}{y})}^{5}$ .

$\sqrt[20]{{(\frac{1}{y})}^{5}}$

Step 4

Rewrite $\sqrt[20]{{(\frac{1}{y})}^{5}}$ as $\sqrt[4]{\sqrt[5]{{(\frac{1}{y})}^{5}}}$ .

$\sqrt[4]{\sqrt[5]{{(\frac{1}{y})}^{5}}}$

Step 5

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

$\sqrt[4]{\frac{1}{y}}$

Step 6

Rewrite $\sqrt[4]{\frac{1}{y}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{y}}$ .

$\frac{\sqrt[4]{1}}{\sqrt[4]{y}}$

Step 7

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[4]{y}}$

Step 8

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

$\frac{1}{\sqrt[4]{y}} \cdot \frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{3}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

Raise $\sqrt[4]{y}$ to the power of $1$ .

$\frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{1} {\sqrt[4]{y}}^{3}}$

Step 9.3

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

$\frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{1 + 3}}$

Step 9.4

Add $1$ and $3$ .

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

Step 9.5

Rewrite ${\sqrt[4]{y}}^{4}$ as $y$ .

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Step 9.5.1

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

$\frac{{\sqrt[4]{y}}^{3}}{{(y^{\frac{1}{4}})}^{4}}$

Step 9.5.2

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

$\frac{{\sqrt[4]{y}}^{3}}{y^{\frac{1}{4} \cdot 4}}$

Step 9.5.3

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

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

Step 9.5.4

Cancel the common factor of $4$ .

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Step 9.5.4.1

Cancel the common factor.

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

Step 9.5.4.2

Rewrite the expression.

$\frac{{\sqrt[4]{y}}^{3}}{y^{1}}$

Step 9.5.5

Simplify.

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

Step 10

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

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