Algebra Examples

$\sqrt{3 v} (2 \sqrt{15 v} - \sqrt{6})$

Step 1

Apply the distributive property.

$\sqrt{3 v} (2 \sqrt{15 v}) + \sqrt{3 v} (- \sqrt{6})$

Step 2

Rewrite using the commutative property of multiplication.

$2 \sqrt{3 v} \sqrt{15 v} + \sqrt{3 v} (- \sqrt{6})$

Step 3

Multiply $\sqrt{3 v} (- \sqrt{6})$ .

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Step 3.1

Combine using the product rule for radicals.

$2 \sqrt{3 v} \sqrt{15 v} - \sqrt{3 v \cdot 6}$

Step 3.2

Multiply $6$ by $3$ .

$2 \sqrt{3 v} \sqrt{15 v} - \sqrt{18 v}$

Step 4

Simplify each term.

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Step 4.1

Multiply $2 \sqrt{3 v} \sqrt{15 v}$ .

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Step 4.1.1

Combine using the product rule for radicals.

$2 \sqrt{15 v (3 v)} - \sqrt{18 v}$

Step 4.1.2

Multiply $3$ by $15$ .

$2 \sqrt{45 v \cdot v} - \sqrt{18 v}$

Step 4.1.3

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

$2 \sqrt{45 (v^{1} v)} - \sqrt{18 v}$

Step 4.1.4

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

$2 \sqrt{45 (v^{1} v^{1})} - \sqrt{18 v}$

Step 4.1.5

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

$2 \sqrt{45 v^{1 + 1}} - \sqrt{18 v}$

Step 4.1.6

Add $1$ and $1$ .

$2 \sqrt{45 v^{2}} - \sqrt{18 v}$

Step 4.2

Rewrite $45 v^{2}$ as ${(3 v)}^{2} \cdot 5$ .

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Step 4.2.1

Factor $9$ out of $45$ .

$2 \sqrt{9 (5) v^{2}} - \sqrt{18 v}$

Step 4.2.2

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

$2 \sqrt{3^{2} \cdot 5 v^{2}} - \sqrt{18 v}$

Step 4.2.3

Move $5$ .

$2 \sqrt{3^{2} v^{2} \cdot 5} - \sqrt{18 v}$

Step 4.2.4

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

$2 \sqrt{{(3 v)}^{2} \cdot 5} - \sqrt{18 v}$

Step 4.3

Pull terms out from under the radical.

$2 (| 3 v | \sqrt{5}) - \sqrt{18 v}$

Step 4.4

Remove non-negative terms from the absolute value.

$2 (3 | v | \sqrt{5}) - \sqrt{18 v}$

Step 4.5

Multiply $3$ by $2$ .

$6 (| v | \sqrt{5}) - \sqrt{18 v}$

Step 4.6

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

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Step 4.6.1

Factor $9$ out of $18$ .

$6 | v | \sqrt{5} - \sqrt{9 (2) v}$

Step 4.6.2

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

$6 | v | \sqrt{5} - \sqrt{3^{2} \cdot 2 v}$

Step 4.6.3

Add parentheses.

$6 | v | \sqrt{5} - \sqrt{3^{2} \cdot (2 v)}$

Step 4.7

Pull terms out from under the radical.

$6 | v | \sqrt{5} - (| 3 | \sqrt{2 v})$

Step 4.8

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$6 | v | \sqrt{5} - (3 \sqrt{2 v})$

Step 4.9

Multiply $3$ by $- 1$ .

$6 | v | \sqrt{5} - 3 \sqrt{2 v}$