Algebra Examples

$(\sqrt[3]{4 v^{7}}) (\sqrt[3]{80 v^{5}})$

Step 1

Rewrite $4 v^{7}$ as ${(v^{2})}^{3} \cdot (4 v)$ .

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Step 1.1

Factor out $v^{6}$ .

$\sqrt[3]{4 (v^{6} v)} \sqrt[3]{80 v^{5}}$

Step 1.2

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

$\sqrt[3]{4 ({(v^{2})}^{3} v)} \sqrt[3]{80 v^{5}}$

Step 1.3

Reorder $4$ and ${(v^{2})}^{3}$ .

$\sqrt[3]{{(v^{2})}^{3} \cdot 4 v} \sqrt[3]{80 v^{5}}$

Step 1.4

Add parentheses.

$\sqrt[3]{{(v^{2})}^{3} \cdot (4 v)} \sqrt[3]{80 v^{5}}$

Step 2

Pull terms out from under the radical.

$v^{2} \sqrt[3]{4 v} \sqrt[3]{80 v^{5}}$

Step 3

Rewrite $80 v^{5}$ as ${(2 v)}^{3} \cdot (10 v^{2})$ .

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Step 3.1

Factor $8$ out of $80$ .

$v^{2} \sqrt[3]{4 v} \sqrt[3]{8 (10) v^{5}}$

Step 3.2

Rewrite $8$ as $2^{3}$ .

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

Step 3.3

Factor out $v^{3}$ .

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

Step 3.4

Move $10$ .

$v^{2} \sqrt[3]{4 v} \sqrt[3]{2^{3} v^{3} \cdot 10 v^{2}}$

Step 3.5

Rewrite $2^{3} v^{3}$ as ${(2 v)}^{3}$ .

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Move $v$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

$2 v^{1 + 2} \sqrt[3]{4 v} \sqrt[3]{10 v^{2}}$

Step 6.3

Add $1$ and $2$ .

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

Step 7

Multiply $2 v^{3} \sqrt[3]{4 v} \sqrt[3]{10 v^{2}}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $4$ by $10$ .

$2 v^{3} \sqrt[3]{40 v^{2} v}$

Step 7.3

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

$2 v^{3} \sqrt[3]{40 (v^{1} v^{2})}$

Step 7.4

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

$2 v^{3} \sqrt[3]{40 v^{1 + 2}}$

Step 7.5

Add $1$ and $2$ .

$2 v^{3} \sqrt[3]{40 v^{3}}$

Step 8

Rewrite $40 v^{3}$ as ${(2 v)}^{3} \cdot 5$ .

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Step 8.1

Factor $8$ out of $40$ .

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

Step 8.2

Rewrite $8$ as $2^{3}$ .

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

Step 8.3

Move $5$ .

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

Step 8.4

Rewrite $2^{3} v^{3}$ as ${(2 v)}^{3}$ .

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $v^{3}$ by $v$ by adding the exponents.

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Step 10.1

Move $v$ .

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

Step 10.2

Multiply $v$ by $v^{3}$ .

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Step 10.2.1

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

$2 (v^{1} v^{3}) (2 \sqrt[3]{5})$

Step 10.2.2

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

$2 v^{1 + 3} (2 \sqrt[3]{5})$

Step 10.3

Add $1$ and $3$ .

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

Step 11

Multiply $2$ by $2$ .

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