Algebra Examples

$3 \sqrt{18 w^{7}} \cdot 10 \sqrt{4 w^{9}}$

Step 1

Rewrite $18 w^{7}$ as ${(3 w^{3})}^{2} \cdot (2 w)$ .

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Step 1.1

Factor $9$ out of $18$ .

$3 \sqrt{9 (2) w^{7}} \cdot (10 \sqrt{4 w^{9}})$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$3 \sqrt{3^{2} \cdot 2 w^{7}} \cdot (10 \sqrt{4 w^{9}})$

Step 1.3

Factor out $w^{6}$ .

$3 \sqrt{3^{2} \cdot 2 (w^{6} w)} \cdot (10 \sqrt{4 w^{9}})$

Step 1.4

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

$3 \sqrt{3^{2} \cdot 2 ({(w^{3})}^{2} w)} \cdot (10 \sqrt{4 w^{9}})$

Step 1.5

Move $2$ .

$3 \sqrt{3^{2} {(w^{3})}^{2} \cdot 2 w} \cdot (10 \sqrt{4 w^{9}})$

Step 1.6

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

$3 \sqrt{{(3 w^{3})}^{2} \cdot 2 w} \cdot (10 \sqrt{4 w^{9}})$

Step 1.7

Add parentheses.

$3 \sqrt{{(3 w^{3})}^{2} \cdot (2 w)} \cdot (10 \sqrt{4 w^{9}})$

Step 2

Pull terms out from under the radical.

$3 (3 w^{3} \sqrt{2 w}) \cdot (10 \sqrt{4 w^{9}})$

Step 3

Multiply $3$ by $3$ .

$9 (w^{3} \sqrt{2 w}) \cdot (10 \sqrt{4 w^{9}})$

Step 4

Rewrite $4 w^{9}$ as ${(2 w^{4})}^{2} w$ .

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Step 4.1

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

$9 w^{3} \sqrt{2 w} \cdot (10 \sqrt{2^{2} w^{9}})$

Step 4.2

Factor out $w^{8}$ .

$9 w^{3} \sqrt{2 w} \cdot (10 \sqrt{2^{2} (w^{8} w)})$

Step 4.3

Rewrite $w^{8}$ as ${(w^{4})}^{2}$ .

$9 w^{3} \sqrt{2 w} \cdot (10 \sqrt{2^{2} ({(w^{4})}^{2} w)})$

Step 4.4

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

$9 w^{3} \sqrt{2 w} \cdot (10 \sqrt{{(2 w^{4})}^{2} w})$

Step 5

Pull terms out from under the radical.

$9 w^{3} \sqrt{2 w} \cdot (10 (2 w^{4} \sqrt{w}))$

Step 6

Multiply $w^{3}$ by $w^{4}$ by adding the exponents.

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Step 6.1

Move $w^{4}$ .

$9 (w^{4} w^{3}) \sqrt{2 w} \cdot (10 (2 \sqrt{w}))$

Step 6.2

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

$9 w^{4 + 3} \sqrt{2 w} \cdot (10 (2 \sqrt{w}))$

Step 6.3

Add $4$ and $3$ .

$9 w^{7} \sqrt{2 w} \cdot (10 (2 \sqrt{w}))$

Step 7

Multiply $2$ by $10$ .

$9 w^{7} \sqrt{2 w} \cdot (20 \sqrt{w})$

Step 8

Multiply $9 w^{7} \sqrt{2 w} (20 \sqrt{w})$ .

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Step 8.1

Multiply $20$ by $9$ .

$180 w^{7} \sqrt{2 w} \sqrt{w}$

Step 8.2

Combine using the product rule for radicals.

$180 w^{7} \sqrt{w (2 w)}$

Step 8.3

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

$180 w^{7} \sqrt{2 (w^{1} w)}$

Step 8.4

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

$180 w^{7} \sqrt{2 (w^{1} w^{1})}$

Step 8.5

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

$180 w^{7} \sqrt{2 w^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$180 w^{7} \sqrt{2 w^{2}}$

Step 9

Reorder $2$ and $w^{2}$ .

$180 w^{7} \sqrt{w^{2} \cdot 2}$

Step 10

Pull terms out from under the radical.

$180 w^{7} (w \sqrt{2})$

Step 11

Multiply $w^{7}$ by $w$ by adding the exponents.

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Step 11.1

Move $w$ .

$180 (w \cdot w^{7}) \sqrt{2}$

Step 11.2

Multiply $w$ by $w^{7}$ .

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Step 11.2.1

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

$180 (w^{1} w^{7}) \sqrt{2}$

Step 11.2.2

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

$180 w^{1 + 7} \sqrt{2}$

Step 11.3

Add $1$ and $7$ .

$180 w^{8} \sqrt{2}$