Simplify fifth root of 2pq^6*2 square root of 2p^3q

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

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

Factor out $q^{5}$ .

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

Move $p$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

Factor out $p^{2}$ .

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

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

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

Add parentheses.

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

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

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

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

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

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

$2 q (p ({(2 p q)}^{\frac{2}{10}} \sqrt{2 p q}))$

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $2$ and $5$ .

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

Step 7

Apply the product rule to $2 p q$ .

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

Step 8

Apply the product rule to $2 p$ .

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

Step 9

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

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

Step 10

Reorder factors in $2 q (p \sqrt[10]{128 p^{7} q^{7}})$ .

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

Simplify fifth root of 2pq^6*2 square root of 2p^3q

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