Pre-Algebra Examples

$\sqrt[3]{y^{7}} \sqrt[9]{y^{6}}$

Step 1

Rewrite $y^{7}$ as ${(y^{2})}^{3} y$ .

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Step 1.1

Factor out $y^{6}$ .

$\sqrt[3]{y^{6} y} \sqrt[9]{y^{6}}$

Step 1.2

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$\sqrt[3]{{(y^{2})}^{3} y} \sqrt[9]{y^{6}}$

Step 2

Pull terms out from under the radical.

$y^{2} \sqrt[3]{y} \sqrt[9]{y^{6}}$

Step 3

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$y^{2} \sqrt[3]{y} \sqrt[9]{{(y^{2})}^{3}}$

Step 4

Rewrite $\sqrt[9]{{(y^{2})}^{3}}$ as $\sqrt[3]{\sqrt[3]{{(y^{2})}^{3}}}$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.2

Add $2$ and $1$ .

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

Step 7

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

$y^{2} y$

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

$y^{2} y^{1}$

Step 8.1.2

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

$y^{2 + 1}$

Step 8.2

Add $2$ and $1$ .

$y^{3}$