Simplify ( cube root of 10v^8)( cube root of 12v^4)

$(\sqrt[3]{10 v^{8}}) (\sqrt[3]{12 v^{4}})$

Step 1

Rewrite $10 v^{8}$ as ${(v^{2})}^{3} \cdot (10 v^{2})$ .

Factor out $v^{6}$ .

$\sqrt[3]{10 (v^{6} v^{2})} \sqrt[3]{12 v^{4}}$

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

$\sqrt[3]{10 ({(v^{2})}^{3} v^{2})} \sqrt[3]{12 v^{4}}$

Reorder $10$ and ${(v^{2})}^{3}$ .

$\sqrt[3]{{(v^{2})}^{3} \cdot 10 v^{2}} \sqrt[3]{12 v^{4}}$

Add parentheses.

$\sqrt[3]{{(v^{2})}^{3} \cdot (10 v^{2})} \sqrt[3]{12 v^{4}}$

Step 2

Pull terms out from under the radical.

$v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v^{4}}$

Step 3

Rewrite $12 v^{4}$ as $v^{3} \cdot (12 v)$ .

Factor out $v^{3}$ .

$v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{12 (v^{3} v)}$

Reorder $12$ and $v^{3}$ .

$v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{v^{3} \cdot 12 v}$

Add parentheses.

$v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{v^{3} \cdot (12 v)}$

Step 4

Pull terms out from under the radical.

$v^{2} \sqrt[3]{10 v^{2}} (v \sqrt[3]{12 v})$

Step 5

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

Move $v$ .

$v \cdot v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v}$

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

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

$v^{1} v^{2} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v}$

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

$v^{1 + 2} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v}$

Add $1$ and $2$ .

$v^{3} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v}$

Step 6

Multiply $v^{3} \sqrt[3]{10 v^{2}} \sqrt[3]{12 v}$ .

Combine using the product rule for radicals.

$v^{3} \sqrt[3]{12 v (10 v^{2})}$

Multiply $10$ by $12$ .

$v^{3} \sqrt[3]{120 v \cdot v^{2}}$

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

Move $v^{2}$ .

$v^{3} \sqrt[3]{120 (v^{2} v)}$

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

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

$v^{3} \sqrt[3]{120 (v^{2} v^{1})}$

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

$v^{3} \sqrt[3]{120 v^{2 + 1}}$

Add $2$ and $1$ .

$v^{3} \sqrt[3]{120 v^{3}}$

Step 7

Rewrite $120 v^{3}$ as ${(2 v)}^{3} \cdot 15$ .

Factor $8$ out of $120$ .

$v^{3} \sqrt[3]{8 (15) v^{3}}$

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

$v^{3} \sqrt[3]{2^{3} \cdot 15 v^{3}}$

Move $15$ .

$v^{3} \sqrt[3]{2^{3} v^{3} \cdot 15}$

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

$v^{3} \sqrt[3]{{(2 v)}^{3} \cdot 15}$

Step 8

Pull terms out from under the radical.

$v^{3} (2 v \sqrt[3]{15})$

Step 9

Rewrite using the commutative property of multiplication.

$2 \sqrt[3]{15} v^{3} v$

Step 10

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

Move $v$ .

$2 \sqrt[3]{15} (v \cdot v^{3})$

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

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

$2 \sqrt[3]{15} (v^{1} v^{3})$

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

$2 \sqrt[3]{15} v^{1 + 3}$

Add $1$ and $3$ .

$2 \sqrt[3]{15} v^{4}$

Simplify ( cube root of 10v^8)( cube root of 12v^4)

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