Algebra Examples

$\sqrt[5]{p^{11}} \sqrt{p^{7}}$

Step 1

Rewrite $p^{11}$ as ${(p^{2})}^{5} p$ .

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Step 1.1

Factor out $p^{10}$ .

$\sqrt[5]{p^{10} p} \sqrt{p^{7}}$

Step 1.2

Rewrite $p^{10}$ as ${(p^{2})}^{5}$ .

$\sqrt[5]{{(p^{2})}^{5} p} \sqrt{p^{7}}$

Step 2

Pull terms out from under the radical.

$p^{2} \sqrt[5]{p} \sqrt{p^{7}}$

Step 3

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

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Step 3.1

Factor out $p^{6}$ .

$p^{2} \sqrt[5]{p} \sqrt{p^{6} p}$

Step 3.2

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

$p^{2} \sqrt[5]{p} \sqrt{{(p^{3})}^{2} p}$

Step 4

Pull terms out from under the radical.

$p^{2} \sqrt[5]{p} (p^{3} \sqrt{p})$

Step 5

Multiply $p^{2}$ by $p^{3}$ by adding the exponents.

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Step 5.1

Move $p^{3}$ .

$p^{3} p^{2} \sqrt[5]{p} \sqrt{p}$

Step 5.2

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

$p^{3 + 2} \sqrt[5]{p} \sqrt{p}$

Step 5.3

Add $3$ and $2$ .

$p^{5} \sqrt[5]{p} \sqrt{p}$

Step 6

Multiply $p^{5} \sqrt[5]{p} \sqrt{p}$ .

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Step 6.1

Rewrite the expression using the least common index of $10$ .

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Step 6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{p}$ as $p^{\frac{1}{2}}$ .

$p^{5} (p^{\frac{1}{2}} \sqrt[5]{p})$

Step 6.1.2

Rewrite $p^{\frac{1}{2}}$ as $p^{\frac{5}{10}}$ .

$p^{5} (p^{\frac{5}{10}} \sqrt[5]{p})$

Step 6.1.3

Rewrite $p^{\frac{5}{10}}$ as $\sqrt[10]{p^{5}}$ .

$p^{5} (\sqrt[10]{p^{5}} \sqrt[5]{p})$

Step 6.1.4

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

$p^{5} (\sqrt[10]{p^{5}} p^{\frac{1}{5}})$

Step 6.1.5

Rewrite $p^{\frac{1}{5}}$ as $p^{\frac{2}{10}}$ .

$p^{5} (\sqrt[10]{p^{5}} p^{\frac{2}{10}})$

Step 6.1.6

Rewrite $p^{\frac{2}{10}}$ as $\sqrt[10]{p^{2}}$ .

$p^{5} (\sqrt[10]{p^{5}} \sqrt[10]{p^{2}})$

Step 6.2

Combine using the product rule for radicals.

$p^{5} \sqrt[10]{p^{5} p^{2}}$

Step 6.3

Multiply $p^{5}$ by $p^{2}$ 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.

$p^{5} \sqrt[10]{p^{5 + 2}}$

Step 6.3.2

Add $5$ and $2$ .

$p^{5} \sqrt[10]{p^{7}}$