Algebra Examples

$\sqrt[5]{2 p q^{6}} \cdot 2 \sqrt{2 p^{3} q}$

Step 1

Rewrite $2 p q^{6}$ as $q^{5} \cdot (2 p q)$ .

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Step 1.1

Factor out $q^{5}$ .

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

Step 1.2

Move $p$ .

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

Step 1.3

Reorder $2$ and $q^{5}$ .

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

Step 1.4

Add parentheses.

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

Step 1.5

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Move $2$ to the left of $q \sqrt[5]{2 p q}$ .

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

Step 4

Rewrite $2 p^{3} q$ as $p^{2} \cdot (2 p q)$ .

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Step 4.1

Factor out $p^{2}$ .

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

Step 4.2

Reorder $2$ and $p^{2}$ .

$2 q \sqrt[5]{2 p q} \sqrt{p^{2} \cdot 2 p q}$

Step 4.3

Add parentheses.

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

Step 4.4

Add parentheses.

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

Step 5

Pull terms out from under the radical.

$2 q \sqrt[5]{2 p q} (| p | \sqrt{2 p q})$

Step 6

Multiply $2 q \sqrt[5]{2 p q} (| p | \sqrt{2 p q})$ .

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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 $\sqrt{2 p q}$ as ${(2 p q)}^{\frac{1}{2}}$ .

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

Step 6.1.2

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

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

Step 6.1.3

Rewrite ${(2 p q)}^{\frac{5}{10}}$ as $\sqrt[10]{{(2 p q)}^{5}}$ .

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

Step 6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 p q}$ as ${(2 p q)}^{\frac{1}{5}}$ .

$2 q (\sqrt[10]{{(2 p q)}^{5}} {(2 p q)}^{\frac{1}{5}}) | p |$

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.2

Combine using the product rule for radicals.

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

Step 6.3

Multiply ${(2 p q)}^{5}$ by ${(2 p q)}^{2}$ 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.

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

Step 6.3.2

Add $5$ and $2$ .

$2 q \sqrt[10]{{(2 p q)}^{7}} | p |$

Step 7

Apply the product rule to $2 p q$ .

$2 q \sqrt[10]{{(2 p)}^{7} q^{7}} | p |$

Step 8

Apply the product rule to $2 p$ .

$2 q \sqrt[10]{2^{7} p^{7} q^{7}} | p |$

Step 9

Raise $2$ to the power of $7$ .

$2 q \sqrt[10]{128 p^{7} q^{7}} | p |$