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

Remove non-negative terms from the absolute value.

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

Step 4

Multiply $3$ by $3$ .

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

Step 5

Multiply $10$ by $9$ .

$90 | w^{3} | \sqrt{2 w} \sqrt{4 w^{9}}$

Step 6

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

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Step 6.1

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

$90 | w^{3} | \sqrt{2 w} \sqrt{2^{2} w^{9}}$

Step 6.2

Factor out $w^{8}$ .

$90 | w^{3} | \sqrt{2 w} \sqrt{2^{2} (w^{8} w)}$

Step 6.3

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

$90 | w^{3} | \sqrt{2 w} \sqrt{2^{2} ({(w^{4})}^{2} w)}$

Step 6.4

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

$90 | w^{3} | \sqrt{2 w} \sqrt{{(2 w^{4})}^{2} w}$

Step 7

Pull terms out from under the radical.

$90 | w^{3} | \sqrt{2 w} (| 2 w^{4} | \sqrt{w})$

Step 8

Remove non-negative terms from the absolute value.

$90 | w^{3} | \sqrt{2 w} (2 w^{4} \sqrt{w})$

Step 9

Multiply $90 | w^{3} | \sqrt{2 w} (2 w^{4} \sqrt{w})$ .

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Step 9.1

Multiply $2$ by $90$ .

$180 | w^{3} | \sqrt{2 w} (w^{4} \sqrt{w})$

Step 9.2

Combine using the product rule for radicals.

$180 | w^{3} | (w^{4} \sqrt{2 w \cdot w})$

Step 9.3

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

$180 | w^{3} | (w^{4} \sqrt{2 (w^{1} w)})$

Step 9.4

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

$180 | w^{3} | (w^{4} \sqrt{2 (w^{1} w^{1})})$

Step 9.5

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

$180 | w^{3} | (w^{4} \sqrt{2 w^{1 + 1}})$

Step 9.6

Add $1$ and $1$ .

$180 | w^{3} | (w^{4} \sqrt{2 w^{2}})$

Step 10

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

$180 | w^{3} | (w^{4} \sqrt{w^{2} \cdot 2})$

Step 11

Pull terms out from under the radical.

$180 | w^{3} | (w^{4} (| w | \sqrt{2}))$

Step 12

Rewrite using the commutative property of multiplication.

$180 | w^{3} | (\sqrt{2} w^{4} | w |)$

Step 13

Multiply $180 | w^{3} | (\sqrt{2} w^{4} | w |)$ .

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Step 13.1

To multiply absolute values, multiply the terms inside each absolute value.

$180 (\sqrt{2} w^{4} | w^{3} w |)$

Step 13.2

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

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Step 13.2.1

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

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Step 13.2.1.1

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

$180 (\sqrt{2} w^{4} | w^{3} w^{1} |)$

Step 13.2.1.2

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

$180 (\sqrt{2} w^{4} | w^{3 + 1} |)$

Step 13.2.2

Add $3$ and $1$ .

$180 (\sqrt{2} w^{4} | w^{4} |)$

Step 14

Remove non-negative terms from the absolute value.

$180 (\sqrt{2} w^{4} w^{4})$

Step 15

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

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Step 15.1

Move $w^{4}$ .

$180 (\sqrt{2} (w^{4} w^{4}))$

Step 15.2

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

$180 (\sqrt{2} w^{4 + 4})$

Step 15.3

Add $4$ and $4$ .

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

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