Precalculus Examples

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

Step 1

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

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

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

Step 1.1.2

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

$\sqrt[6]{y^{5}} y^{\frac{4}{6}}$

Step 1.1.3

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

$\sqrt[6]{y^{5}} \sqrt[6]{y^{4}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[6]{y^{5} y^{4}}$

Step 1.3

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

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Step 1.3.1

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

$\sqrt[6]{y^{5 + 4}}$

Step 1.3.2

Add $5$ and $4$ .

$\sqrt[6]{y^{9}}$

Step 2

Factor out $y^{6}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

Step 5

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

$y \sqrt{y}$