Trigonometry Examples

$(\sqrt[8]{y^{7}}) (\sqrt[3]{y^{2}})$

Step 1

Multiply $(\sqrt[8]{y^{7}}) (\sqrt[3]{y^{2}})$ .

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Step 1.1

Rewrite the expression using the least common index of $24$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{y^{7}}$ as $y^{\frac{7}{8}}$ .

$y^{\frac{7}{8}} \sqrt[3]{y^{2}}$

Step 1.1.2

Rewrite $y^{\frac{7}{8}}$ as $y^{\frac{21}{24}}$ .

$y^{\frac{21}{24}} \sqrt[3]{y^{2}}$

Step 1.1.3

Rewrite $y^{\frac{21}{24}}$ as $\sqrt[24]{y^{21}}$ .

$\sqrt[24]{y^{21}} \sqrt[3]{y^{2}}$

Step 1.1.4

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

$\sqrt[24]{y^{21}} y^{\frac{2}{3}}$

Step 1.1.5

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{16}{24}}$ .

$\sqrt[24]{y^{21}} y^{\frac{16}{24}}$

Step 1.1.6

Rewrite $y^{\frac{16}{24}}$ as $\sqrt[24]{y^{16}}$ .

$\sqrt[24]{y^{21}} \sqrt[24]{y^{16}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[24]{y^{21} y^{16}}$

Step 1.3

Multiply $y^{21}$ by $y^{16}$ 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.

$\sqrt[24]{y^{21 + 16}}$

Step 1.3.2

Add $21$ and $16$ .

$\sqrt[24]{y^{37}}$

Step 2

Factor out $y^{24}$ .

$\sqrt[24]{y^{24} y^{13}}$

Step 3

Pull terms out from under the radical.

$y \sqrt[24]{y^{13}}$