Basic Math Examples

$\sqrt[12]{y^{7}} \cdot \sqrt[3]{y^{5}}$

Step 1

Factor out $y^{3}$ .

$\sqrt[12]{y^{7}} \cdot \sqrt[3]{y^{3} y^{2}}$

Step 2

Pull terms out from under the radical.

$\sqrt[12]{y^{7}} \cdot (y \sqrt[3]{y^{2}})$

Step 3

Multiply $\sqrt[12]{y^{7}} (y \sqrt[3]{y^{2}})$ .

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Step 3.1

Rewrite the expression using the least common index of $12$ .

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Step 3.1.1

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

$y (\sqrt[12]{y^{7}} y^{\frac{2}{3}})$

Step 3.1.2

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{8}{12}}$ .

$y (\sqrt[12]{y^{7}} y^{\frac{8}{12}})$

Step 3.1.3

Rewrite $y^{\frac{8}{12}}$ as $\sqrt[12]{y^{8}}$ .

$y (\sqrt[12]{y^{7}} \sqrt[12]{y^{8}})$

Step 3.2

Combine using the product rule for radicals.

$y \sqrt[12]{y^{7} y^{8}}$

Step 3.3

Multiply $y^{7}$ by $y^{8}$ 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.

$y \sqrt[12]{y^{7 + 8}}$

Step 3.3.2

Add $7$ and $8$ .

$y \sqrt[12]{y^{15}}$

Step 4

Factor out $y^{12}$ .

$y \sqrt[12]{y^{12} y^{3}}$

Step 5

Pull terms out from under the radical.

$y (y \sqrt[12]{y^{3}})$

Step 6

Multiply $y$ by $y$ by adding the exponents.

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Step 6.1

Move $y$ .

$y \cdot y \sqrt[12]{y^{3}}$

Step 6.2

Multiply $y$ by $y$ .

$y^{2} \sqrt[12]{y^{3}}$

Step 7

Rewrite $\sqrt[12]{y^{3}}$ as $\sqrt[4]{\sqrt[3]{y^{3}}}$ .

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

Step 8

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

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