Basic Math Examples

$\frac{\sqrt[3]{y^{2}}}{\sqrt[5]{y^{2}}}$

Step 1

Multiply $\frac{\sqrt[3]{y^{2}}}{\sqrt[5]{y^{2}}}$ by $\frac{{\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{4}}$ .

$\frac{\sqrt[3]{y^{2}}}{\sqrt[5]{y^{2}}} \cdot \frac{{\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{4}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[3]{y^{2}}}{\sqrt[5]{y^{2}}}$ by $\frac{{\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{4}}$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{\sqrt[5]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}$

Step 2.2

Raise $\sqrt[5]{y^{2}}$ to the power of $1$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{1} {\sqrt[5]{y^{2}}}^{4}}$

Step 2.3

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

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{1 + 4}}$

Step 2.4

Add $1$ and $4$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{{\sqrt[5]{y^{2}}}^{5}}$

Step 2.5

Rewrite ${\sqrt[5]{y^{2}}}^{5}$ as $y^{2}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{2}}$ as $y^{\frac{2}{5}}$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{{(y^{\frac{2}{5}})}^{5}}$

Step 2.5.2

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

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{2}{5} \cdot 5}}$

Step 2.5.3

Combine $\frac{2}{5}$ and $5$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{2 \cdot 5}{5}}}$

Step 2.5.4

Multiply $2$ by $5$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{10}{5}}}$

Step 2.5.5

Cancel the common factor of $10$ and $5$ .

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Step 2.5.5.1

Factor $5$ out of $10$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{5 \cdot 2}{5}}}$

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $5$ out of $5$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{5 \cdot 2}{5 (1)}}}$

Step 2.5.5.2.2

Cancel the common factor.

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{5 \cdot 2}{5 \cdot 1}}}$

Step 2.5.5.2.3

Rewrite the expression.

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{\frac{2}{1}}}$

Step 2.5.5.2.4

Divide $2$ by $1$ .

$\frac{\sqrt[3]{y^{2}} {\sqrt[5]{y^{2}}}^{4}}{y^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{y^{2}}}^{4}$ as $\sqrt[5]{{(y^{2})}^{4}}$ .

$\frac{\sqrt[3]{y^{2}} \sqrt[5]{{(y^{2})}^{4}}}{y^{2}}$

Step 3.2

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 3.2.1

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

$\frac{\sqrt[3]{y^{2}} \sqrt[5]{y^{2 \cdot 4}}}{y^{2}}$

Step 3.2.2

Multiply $2$ by $4$ .

$\frac{\sqrt[3]{y^{2}} \sqrt[5]{y^{8}}}{y^{2}}$

Step 3.3

Factor out $y^{5}$ .

$\frac{\sqrt[3]{y^{2}} \sqrt[5]{y^{5} y^{3}}}{y^{2}}$

Step 3.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{y^{2}} y \sqrt[5]{y^{3}}}{y^{2}}$

Step 3.5

Combine exponents.

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Step 3.5.1

Rewrite the expression using the least common index of $15$ .

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Step 3.5.1.1

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

$\frac{y (y^{\frac{2}{3}} \sqrt[5]{y^{3}})}{y^{2}}$

Step 3.5.1.2

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{10}{15}}$ .

$\frac{y (y^{\frac{10}{15}} \sqrt[5]{y^{3}})}{y^{2}}$

Step 3.5.1.3

Rewrite $y^{\frac{10}{15}}$ as $\sqrt[15]{y^{10}}$ .

$\frac{y (\sqrt[15]{y^{10}} \sqrt[5]{y^{3}})}{y^{2}}$

Step 3.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{3}}$ as $y^{\frac{3}{5}}$ .

$\frac{y (\sqrt[15]{y^{10}} y^{\frac{3}{5}})}{y^{2}}$

Step 3.5.1.5

Rewrite $y^{\frac{3}{5}}$ as $y^{\frac{9}{15}}$ .

$\frac{y (\sqrt[15]{y^{10}} y^{\frac{9}{15}})}{y^{2}}$

Step 3.5.1.6

Rewrite $y^{\frac{9}{15}}$ as $\sqrt[15]{y^{9}}$ .

$\frac{y (\sqrt[15]{y^{10}} \sqrt[15]{y^{9}})}{y^{2}}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{y \sqrt[15]{y^{10} y^{9}}}{y^{2}}$

Step 3.5.3

Multiply $y^{10}$ by $y^{9}$ by adding the exponents.

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Step 3.5.3.1

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

$\frac{y \sqrt[15]{y^{10 + 9}}}{y^{2}}$

Step 3.5.3.2

Add $10$ and $9$ .

$\frac{y \sqrt[15]{y^{19}}}{y^{2}}$

Step 3.6

Factor out $y^{15}$ .

$\frac{y \sqrt[15]{y^{15} y^{4}}}{y^{2}}$

Step 3.7

Pull terms out from under the radical.

$\frac{y (y \sqrt[15]{y^{4}})}{y^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{y^{1} y \sqrt[15]{y^{4}}}{y^{2}}$

Step 4.2

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

$\frac{y^{1} y^{1} \sqrt[15]{y^{4}}}{y^{2}}$

Step 4.3

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

$\frac{y^{1 + 1} \sqrt[15]{y^{4}}}{y^{2}}$

Step 4.4

Add $1$ and $1$ .

$\frac{y^{2} \sqrt[15]{y^{4}}}{y^{2}}$

Step 5

Cancel the common factor of $y^{2}$ .

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Step 5.1

Cancel the common factor.

$\frac{y^{2} \sqrt[15]{y^{4}}}{y^{2}}$

Step 5.2

Divide $\sqrt[15]{y^{4}}$ by $1$ .

$\sqrt[15]{y^{4}}$