Simplify cube root of 6w^7* cube root of 4w^5

$\sqrt[3]{6 w^{7}} \cdot \sqrt[3]{4 w^{5}}$

Step 1

Rewrite $6 w^{7}$ as ${(w^{2})}^{3} \cdot (6 w)$ .

Factor out $w^{6}$ .

$\sqrt[3]{6 (w^{6} w)} \cdot \sqrt[3]{4 w^{5}}$

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

$\sqrt[3]{6 ({(w^{2})}^{3} w)} \cdot \sqrt[3]{4 w^{5}}$

Reorder $6$ and ${(w^{2})}^{3}$ .

$\sqrt[3]{{(w^{2})}^{3} \cdot 6 w} \cdot \sqrt[3]{4 w^{5}}$

Add parentheses.

$\sqrt[3]{{(w^{2})}^{3} \cdot (6 w)} \cdot \sqrt[3]{4 w^{5}}$

Step 2

Pull terms out from under the radical.

$w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{4 w^{5}}$

Step 3

Rewrite $4 w^{5}$ as $w^{3} \cdot (4 w^{2})$ .

Factor out $w^{3}$ .

$w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{4 (w^{3} w^{2})}$

Reorder $4$ and $w^{3}$ .

$w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{w^{3} \cdot 4 w^{2}}$

Add parentheses.

$w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{w^{3} \cdot (4 w^{2})}$

Step 4

Pull terms out from under the radical.

$w^{2} \sqrt[3]{6 w} \cdot (w \sqrt[3]{4 w^{2}})$

Step 5

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

Move $w$ .

$w \cdot w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{4 w^{2}}$

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

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

$w^{1} w^{2} \sqrt[3]{6 w} \cdot \sqrt[3]{4 w^{2}}$

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

$w^{1 + 2} \sqrt[3]{6 w} \cdot \sqrt[3]{4 w^{2}}$

Add $1$ and $2$ .

$w^{3} \sqrt[3]{6 w} \cdot \sqrt[3]{4 w^{2}}$

Step 6

Multiply $w^{3} \sqrt[3]{6 w} \sqrt[3]{4 w^{2}}$ .

Combine using the product rule for radicals.

$w^{3} \sqrt[3]{4 w^{2} (6 w)}$

Multiply $6$ by $4$ .

$w^{3} \sqrt[3]{24 w^{2} w}$

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

$w^{3} \sqrt[3]{24 (w^{1} w^{2})}$

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

$w^{3} \sqrt[3]{24 w^{1 + 2}}$

Add $1$ and $2$ .

$w^{3} \sqrt[3]{24 w^{3}}$

Step 7

Rewrite $24 w^{3}$ as ${(2 w)}^{3} \cdot 3$ .

Factor $8$ out of $24$ .

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

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

$w^{3} \sqrt[3]{2^{3} \cdot 3 w^{3}}$

Move $3$ .

$w^{3} \sqrt[3]{2^{3} w^{3} \cdot 3}$

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Rewrite using the commutative property of multiplication.

$2 \sqrt[3]{3} w^{3} w$

Step 10

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

Move $w$ .

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

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

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

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

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

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

Add $1$ and $3$ .

$2 \sqrt[3]{3} w^{4}$

Simplify cube root of 6w^7* cube root of 4w^5

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