Basic Math Examples

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

Step 1

Rewrite $y^{5}$ as ${(y^{2})}^{2} y$ .

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Step 1.1

Factor out $y^{4}$ .

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Rewrite the expression using the least common index of $6$ .

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Step 3.1.1

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

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

Step 3.1.2

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{4}{6}}$ .

$y^{2} (y^{\frac{4}{6}} \sqrt{y})$

Step 3.1.3

Rewrite $y^{\frac{4}{6}}$ as $\sqrt[6]{y^{4}}$ .

$y^{2} (\sqrt[6]{y^{4}} \sqrt{y})$

Step 3.1.4

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

$y^{2} (\sqrt[6]{y^{4}} y^{\frac{1}{2}})$

Step 3.1.5

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{3}{6}}$ .

$y^{2} (\sqrt[6]{y^{4}} y^{\frac{3}{6}})$

Step 3.1.6

Rewrite $y^{\frac{3}{6}}$ as $\sqrt[6]{y^{3}}$ .

$y^{2} (\sqrt[6]{y^{4}} \sqrt[6]{y^{3}})$

Step 3.2

Combine using the product rule for radicals.

$y^{2} \sqrt[6]{y^{4} y^{3}}$

Step 3.3

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 3.3.1

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

$y^{2} \sqrt[6]{y^{4 + 3}}$

Step 3.3.2

Add $4$ and $3$ .

$y^{2} \sqrt[6]{y^{7}}$

Step 4

Factor out $y^{6}$ .

$y^{2} \sqrt[6]{y^{6} y}$

Step 5

Pull terms out from under the radical.

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

Step 6

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 6.1

Move $y$ .

$y \cdot y^{2} \sqrt[6]{y}$

Step 6.2

Multiply $y$ by $y^{2}$ .

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Step 6.2.1

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

$y^{1} y^{2} \sqrt[6]{y}$

Step 6.2.2

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

$y^{1 + 2} \sqrt[6]{y}$

Step 6.3

Add $1$ and $2$ .

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