Algebra Examples

$\sqrt[6]{y} \cdot \sqrt[3]{y} \cdot \sqrt[5]{y^{2}}$

Step 1

Multiply $\sqrt[6]{y} \sqrt[3]{y}$ .

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Step 1.1

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

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Step 1.1.1

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

$\sqrt[6]{y} y^{\frac{1}{3}} \cdot \sqrt[5]{y^{2}}$

Step 1.1.2

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

$\sqrt[6]{y} y^{\frac{2}{6}} \cdot \sqrt[5]{y^{2}}$

Step 1.1.3

Rewrite $y^{\frac{2}{6}}$ as $\sqrt[6]{y^{2}}$ .

$\sqrt[6]{y} \sqrt[6]{y^{2}} \cdot \sqrt[5]{y^{2}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[6]{y \cdot y^{2}} \cdot \sqrt[5]{y^{2}}$

Step 1.3

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

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Step 1.3.1

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

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Step 1.3.1.1

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

$\sqrt[6]{y^{1} y^{2}} \cdot \sqrt[5]{y^{2}}$

Step 1.3.1.2

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

$\sqrt[6]{y^{1 + 2}} \cdot \sqrt[5]{y^{2}}$

Step 1.3.2

Add $1$ and $2$ .

$\sqrt[6]{y^{3}} \cdot \sqrt[5]{y^{2}}$

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Rewrite the expression using the least common index of $10$ .

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Step 4.1.1

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

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

Step 4.1.2

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{5}{10}}$ .

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

Step 4.1.3

Rewrite $y^{\frac{5}{10}}$ as $\sqrt[10]{y^{5}}$ .

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

Step 4.1.4

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

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

Step 4.1.5

Rewrite $y^{\frac{2}{5}}$ as $y^{\frac{4}{10}}$ .

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

Step 4.1.6

Rewrite $y^{\frac{4}{10}}$ as $\sqrt[10]{y^{4}}$ .

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Multiply $y^{5}$ by $y^{4}$ by adding the exponents.

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Step 4.3.1

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

$\sqrt[10]{y^{5 + 4}}$

Step 4.3.2

Add $5$ and $4$ .

$\sqrt[10]{y^{9}}$