Basic Math Examples

$\sqrt[5]{4 u^{2} v^{6}} \cdot \sqrt{8 u^{3} v^{4}}$

Step 1

Rewrite $4 u^{2} v^{6}$ as $v^{5} \cdot (4 u^{2} v)$ .

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Step 1.1

Factor out $v^{5}$ .

$\sqrt[5]{4 u^{2} (v^{5} v)} \cdot \sqrt{8 u^{3} v^{4}}$

Step 1.2

Move $u^{2}$ .

$\sqrt[5]{4 v^{5} u^{2} v} \cdot \sqrt{8 u^{3} v^{4}}$

Step 1.3

Reorder $4$ and $v^{5}$ .

$\sqrt[5]{v^{5} \cdot 4 u^{2} v} \cdot \sqrt{8 u^{3} v^{4}}$

Step 1.4

Add parentheses.

$\sqrt[5]{v^{5} \cdot 4 (u^{2} v)} \cdot \sqrt{8 u^{3} v^{4}}$

Step 1.5

Add parentheses.

$\sqrt[5]{v^{5} \cdot (4 u^{2} v)} \cdot \sqrt{8 u^{3} v^{4}}$

Step 2

Pull terms out from under the radical.

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{8 u^{3} v^{4}}$

Step 3

Rewrite $8 u^{3} v^{4}$ as ${(2 u v^{2})}^{2} \cdot (2 u)$ .

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Step 3.1

Factor $4$ out of $8$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{4 (2) u^{3} v^{4}}$

Step 3.2

Rewrite $4$ as $2^{2}$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{2^{2} \cdot 2 u^{3} v^{4}}$

Step 3.3

Factor out $u^{2}$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{2^{2} \cdot 2 (u^{2} u) v^{4}}$

Step 3.4

Rewrite $v^{4}$ as ${(v^{2})}^{2}$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{2^{2} \cdot 2 (u^{2} u) {(v^{2})}^{2}}$

Step 3.5

Move $u$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{2^{2} \cdot 2 (u^{2}) {(v^{2})}^{2} u}$

Step 3.6

Move $2$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{2^{2} (u^{2}) {(v^{2})}^{2} \cdot 2 u}$

Step 3.7

Rewrite $(2^{2} (u^{2})) {(v^{2})}^{2}$ as ${(2 u v^{2})}^{2}$ .

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{{(2 u v^{2})}^{2} \cdot 2 u}$

Step 3.8

Add parentheses.

$v \sqrt[5]{4 u^{2} v} \cdot \sqrt{{(2 u v^{2})}^{2} \cdot (2 u)}$

Step 4

Pull terms out from under the radical.

$v \sqrt[5]{4 u^{2} v} \cdot (2 u v^{2} \sqrt{2 u})$

Step 5

Rewrite using the commutative property of multiplication.

$2 v \sqrt[5]{4 u^{2} v} (u v^{2} \sqrt{2 u})$

Step 6

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

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Step 6.1

Move $v^{2}$ .

$2 (v^{2} v) \sqrt[5]{4 u^{2} v} (u \sqrt{2 u})$

Step 6.2

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

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Step 6.2.1

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

$2 (v^{2} v^{1}) \sqrt[5]{4 u^{2} v} (u \sqrt{2 u})$

Step 6.2.2

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

$2 v^{2 + 1} \sqrt[5]{4 u^{2} v} (u \sqrt{2 u})$

Step 6.3

Add $2$ and $1$ .

$2 v^{3} \sqrt[5]{4 u^{2} v} (u \sqrt{2 u})$

Step 7

Multiply $2 v^{3} \sqrt[5]{4 u^{2} v} (u \sqrt{2 u})$ .

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Step 7.1

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

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Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 u^{2} v}$ as ${(4 u^{2} v)}^{\frac{1}{5}}$ .

$2 v^{3} (u ({(4 u^{2} v)}^{\frac{1}{5}} \sqrt{2 u}))$

Step 7.1.2

Rewrite ${(4 u^{2} v)}^{\frac{1}{5}}$ as ${(4 u^{2} v)}^{\frac{2}{10}}$ .

$2 v^{3} (u ({(4 u^{2} v)}^{\frac{2}{10}} \sqrt{2 u}))$

Step 7.1.3

Rewrite ${(4 u^{2} v)}^{\frac{2}{10}}$ as $\sqrt[10]{{(4 u^{2} v)}^{2}}$ .

$2 v^{3} (u (\sqrt[10]{{(4 u^{2} v)}^{2}} \sqrt{2 u}))$

Step 7.1.4

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

$2 v^{3} (u (\sqrt[10]{{(4 u^{2} v)}^{2}} {(2 u)}^{\frac{1}{2}}))$

Step 7.1.5

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

$2 v^{3} (u (\sqrt[10]{{(4 u^{2} v)}^{2}} {(2 u)}^{\frac{5}{10}}))$

Step 7.1.6

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

$2 v^{3} (u (\sqrt[10]{{(4 u^{2} v)}^{2}} \sqrt[10]{{(2 u)}^{5}}))$

Step 7.2

Combine using the product rule for radicals.

$2 v^{3} (u \sqrt[10]{{(4 u^{2} v)}^{2} {(2 u)}^{5}})$

Step 7.3

Apply the product rule to $4 u^{2} v$ .

$2 v^{3} (u \sqrt[10]{{(4 u^{2})}^{2} v^{2} {(2 u)}^{5}})$

Step 7.4

Apply the product rule to $2 u$ .

$2 v^{3} (u \sqrt[10]{{(4 u^{2})}^{2} v^{2} \cdot 2^{5} u^{5}})$

Step 7.5

Apply the product rule to $4 u^{2}$ .

$2 v^{3} (u \sqrt[10]{4^{2} {(u^{2})}^{2} v^{2} \cdot 2^{5} u^{5}})$

Step 7.6

Raise $4$ to the power of $2$ .

$2 v^{3} (u \sqrt[10]{16 {(u^{2})}^{2} v^{2} \cdot 2^{5} u^{5}})$

Step 7.7

Multiply the exponents in ${(u^{2})}^{2}$ .

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Step 7.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 v^{3} (u \sqrt[10]{16 u^{2 \cdot 2} v^{2} \cdot 2^{5} u^{5}})$

Step 7.7.2

Multiply $2$ by $2$ .

$2 v^{3} (u \sqrt[10]{16 u^{4} v^{2} \cdot 2^{5} u^{5}})$

Step 7.8

Raise $2$ to the power of $5$ .

$2 v^{3} (u \sqrt[10]{16 u^{4} v^{2} \cdot 32 u^{5}})$

Step 7.9

Combine exponents.

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Step 7.9.1

Multiply $32$ by $16$ .

$2 v^{3} (u \sqrt[10]{512 u^{4} v^{2} u^{5}})$

Step 7.9.2

Multiply $u^{4}$ by $u^{5}$ by adding the exponents.

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Step 7.9.2.1

Move $u^{5}$ .

$2 v^{3} (u \sqrt[10]{512 (u^{5} u^{4}) v^{2}})$

Step 7.9.2.2

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

$2 v^{3} (u \sqrt[10]{512 u^{5 + 4} v^{2}})$

Step 7.9.2.3

Add $5$ and $4$ .

$2 v^{3} (u \sqrt[10]{512 u^{9} v^{2}})$

$2 v^{3} u \sqrt[10]{512 u^{9} v^{2}}$