Algebra Examples

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

Step 1

Rewrite the expression using the least common index of $15$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y}$ as $y^{\frac{1}{5}}$ .

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

Step 1.2

Rewrite $y^{\frac{1}{5}}$ as $y^{\frac{3}{15}}$ .

$y^{\frac{3}{15}} \sqrt[3]{y^{2}}$

Step 1.3

Rewrite $y^{\frac{3}{15}}$ as $\sqrt[15]{y^{3}}$ .

$\sqrt[15]{y^{3}} \sqrt[3]{y^{2}}$

Step 1.4

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

$\sqrt[15]{y^{3}} y^{\frac{2}{3}}$

Step 1.5

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{10}{15}}$ .

$\sqrt[15]{y^{3}} y^{\frac{10}{15}}$

Step 1.6

Rewrite $y^{\frac{10}{15}}$ as $\sqrt[15]{y^{10}}$ .

$\sqrt[15]{y^{3}} \sqrt[15]{y^{10}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[15]{y^{3} y^{10}}$

Step 3

Multiply $y^{3}$ by $y^{10}$ by adding the exponents.

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Step 3.1

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

$\sqrt[15]{y^{3 + 10}}$

Step 3.2

Add $3$ and $10$ .

$\sqrt[15]{y^{13}}$