Algebra Examples

$\sqrt[6]{r^{11}} \sqrt[5]{r^{8}}$

Step 1

Factor out $r^{6}$ .

$\sqrt[6]{r^{6} r^{5}} \sqrt[5]{r^{8}}$

Step 2

Pull terms out from under the radical.

$r \sqrt[6]{r^{5}} \sqrt[5]{r^{8}}$

Step 3

Factor out $r^{5}$ .

$r \sqrt[6]{r^{5}} \sqrt[5]{r^{5} r^{3}}$

Step 4

Pull terms out from under the radical.

$r \sqrt[6]{r^{5}} (r \sqrt[5]{r^{3}})$

Step 5

Multiply $r$ by $r$ by adding the exponents.

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Step 5.1

Move $r$ .

$r \cdot r \sqrt[6]{r^{5}} \sqrt[5]{r^{3}}$

Step 5.2

Multiply $r$ by $r$ .

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

Step 6

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

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Step 6.1

Rewrite the expression using the least common index of $30$ .

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{r^{3}}$ as $r^{\frac{3}{5}}$ .

$r^{2} (r^{\frac{3}{5}} \sqrt[6]{r^{5}})$

Step 6.1.2

Rewrite $r^{\frac{3}{5}}$ as $r^{\frac{18}{30}}$ .

$r^{2} (r^{\frac{18}{30}} \sqrt[6]{r^{5}})$

Step 6.1.3

Rewrite $r^{\frac{18}{30}}$ as $\sqrt[30]{r^{18}}$ .

$r^{2} (\sqrt[30]{r^{18}} \sqrt[6]{r^{5}})$

Step 6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{r^{5}}$ as $r^{\frac{5}{6}}$ .

$r^{2} (\sqrt[30]{r^{18}} r^{\frac{5}{6}})$

Step 6.1.5

Rewrite $r^{\frac{5}{6}}$ as $r^{\frac{25}{30}}$ .

$r^{2} (\sqrt[30]{r^{18}} r^{\frac{25}{30}})$

Step 6.1.6

Rewrite $r^{\frac{25}{30}}$ as $\sqrt[30]{r^{25}}$ .

$r^{2} (\sqrt[30]{r^{18}} \sqrt[30]{r^{25}})$

Step 6.2

Combine using the product rule for radicals.

$r^{2} \sqrt[30]{r^{18} r^{25}}$

Step 6.3

Multiply $r^{18}$ by $r^{25}$ by adding the exponents.

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Step 6.3.1

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

$r^{2} \sqrt[30]{r^{18 + 25}}$

Step 6.3.2

Add $18$ and $25$ .

$r^{2} \sqrt[30]{r^{43}}$

Step 7

Factor out $r^{30}$ .

$r^{2} \sqrt[30]{r^{30} r^{13}}$

Step 8

Pull terms out from under the radical.

$r^{2} (r \sqrt[30]{r^{13}})$

Step 9

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

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Step 9.1

Move $r$ .

$r \cdot r^{2} \sqrt[30]{r^{13}}$

Step 9.2

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

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Step 9.2.1

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

$r^{1} r^{2} \sqrt[30]{r^{13}}$

Step 9.2.2

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

$r^{1 + 2} \sqrt[30]{r^{13}}$

Step 9.3

Add $1$ and $2$ .

$r^{3} \sqrt[30]{r^{13}}$