Simplify 2 fourth root of 8w^3*3 fourth root of 32w^5

$2 \sqrt[4]{8 w^{3}} \cdot 3 \sqrt[4]{32 w^{5}}$

Step 1

Rewrite $32 w^{5}$ as ${(2 w)}^{4} \cdot (2 w)$ .

Factor $16$ out of $32$ .

$2 \sqrt[4]{8 w^{3}} \cdot (3 \sqrt[4]{16 (2) w^{5}})$

Rewrite $16$ as $2^{4}$ .

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

Factor out $w^{4}$ .

$2 \sqrt[4]{8 w^{3}} \cdot (3 \sqrt[4]{2^{4} \cdot 2 (w^{4} w)})$

Move $2$ .

$2 \sqrt[4]{8 w^{3}} \cdot (3 \sqrt[4]{2^{4} w^{4} \cdot 2 w})$

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

$2 \sqrt[4]{8 w^{3}} \cdot (3 \sqrt[4]{{(2 w)}^{4} \cdot 2 w})$

Add parentheses.

$2 \sqrt[4]{8 w^{3}} \cdot (3 \sqrt[4]{{(2 w)}^{4} \cdot (2 w)})$

Step 2

Pull terms out from under the radical.

$2 \sqrt[4]{8 w^{3}} \cdot (3 (2 w \sqrt[4]{2 w}))$

Step 3

Multiply $2$ by $3$ .

$2 \sqrt[4]{8 w^{3}} \cdot (6 (w \sqrt[4]{2 w}))$

Step 4

Multiply $2 \sqrt[4]{8 w^{3}} (6 w \sqrt[4]{2 w})$ .

Multiply $6$ by $2$ .

$12 \sqrt[4]{8 w^{3}} (w \sqrt[4]{2 w})$

Combine using the product rule for radicals.

$12 (w \sqrt[4]{8 w^{3} (2 w)})$

Multiply $2$ by $8$ .

$12 (w \sqrt[4]{16 w^{3} w})$

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

$12 (w \sqrt[4]{16 (w^{1} w^{3})})$

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

$12 (w \sqrt[4]{16 w^{1 + 3}})$

Add $1$ and $3$ .

$12 (w \sqrt[4]{16 w^{4}})$

Step 5

Rewrite $16 w^{4}$ as ${(2 w)}^{4}$ .

$12 (w \sqrt[4]{{(2 w)}^{4}})$

Step 6

Pull terms out from under the radical, assuming positive real numbers.

$12 (w (2 w))$

Step 7

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$12 (w \cdot w \cdot 2)$

Multiply $w$ by $w$ .

$12 (w^{2} \cdot 2)$

Step 8

Move $2$ to the left of $w^{2}$ .

$12 (2 \cdot w^{2})$

Step 9

Multiply $2$ by $12$ .

$24 w^{2}$

Simplify 2 fourth root of 8w^3*3 fourth root of 32w^5

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