Algebra Examples

$\sqrt{24 x} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 1

Rewrite $24 x$ as $2^{2} \cdot (6 x)$ .

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Step 1.1

Factor $4$ out of $24$ .

$\sqrt{4 (6) x} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 1.2

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

$\sqrt{2^{2} \cdot 6 x} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 1.3

Add parentheses.

$\sqrt{2^{2} \cdot (6 x)} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 2

Pull terms out from under the radical.

$| 2 | \sqrt{6 x} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$2 \sqrt{6 x} \cdot \sqrt{18 y} \cdot \sqrt{15 y}$

Step 4

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

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Step 4.1

Factor $9$ out of $18$ .

$2 \sqrt{6 x} \cdot \sqrt{9 (2) y} \cdot \sqrt{15 y}$

Step 4.2

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

$2 \sqrt{6 x} \cdot \sqrt{3^{2} \cdot 2 y} \cdot \sqrt{15 y}$

Step 4.3

Add parentheses.

$2 \sqrt{6 x} \cdot \sqrt{3^{2} \cdot (2 y)} \cdot \sqrt{15 y}$

Step 5

Pull terms out from under the radical.

$2 \sqrt{6 x} \cdot (| 3 | \sqrt{2 y}) \cdot \sqrt{15 y}$

Step 6

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

$2 \sqrt{6 x} \cdot (3 \sqrt{2 y}) \cdot \sqrt{15 y}$

Step 7

Multiply $2 \sqrt{6 x} (3 \sqrt{2 y})$ .

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Step 7.1

Multiply $3$ by $2$ .

$6 \sqrt{6 x} \sqrt{2 y} \cdot \sqrt{15 y}$

Step 7.2

Combine using the product rule for radicals.

$6 \sqrt{2 y (6 x)} \cdot \sqrt{15 y}$

Step 7.3

Multiply $6$ by $2$ .

$6 \sqrt{12 y x} \cdot \sqrt{15 y}$

Step 8

Rewrite $12 y x$ as $2^{2} \cdot (3 y x)$ .

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Step 8.1

Factor $4$ out of $12$ .

$6 \sqrt{4 (3) y x} \cdot \sqrt{15 y}$

Step 8.2

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

$6 \sqrt{2^{2} \cdot 3 y x} \cdot \sqrt{15 y}$

Step 8.3

Add parentheses.

$6 \sqrt{2^{2} \cdot 3 (y x)} \cdot \sqrt{15 y}$

Step 8.4

Add parentheses.

$6 \sqrt{2^{2} \cdot (3 y x)} \cdot \sqrt{15 y}$

Step 9

Pull terms out from under the radical.

$6 (| 2 | \sqrt{3 y x}) \cdot \sqrt{15 y}$

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$6 (2 \sqrt{3 y x}) \cdot \sqrt{15 y}$

Step 11

Multiply $2$ by $6$ .

$12 \sqrt{3 y x} \cdot \sqrt{15 y}$

Step 12

Multiply $12 \sqrt{3 y x} \sqrt{15 y}$ .

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Step 12.1

Combine using the product rule for radicals.

$12 \sqrt{15 y (3 y x)}$

Step 12.2

Multiply $3$ by $15$ .

$12 \sqrt{45 y (y x)}$

Step 12.3

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

$12 \sqrt{45 (y^{1} y) x}$

Step 12.4

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

$12 \sqrt{45 (y^{1} y^{1}) x}$

Step 12.5

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

$12 \sqrt{45 y^{1 + 1} x}$

Step 12.6

Add $1$ and $1$ .

$12 \sqrt{45 y^{2} x}$

Step 13

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

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Step 13.1

Factor $9$ out of $45$ .

$12 \sqrt{9 (5) y^{2} x}$

Step 13.2

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

$12 \sqrt{3^{2} \cdot 5 y^{2} x}$

Step 13.3

Move $5$ .

$12 \sqrt{3^{2} y^{2} \cdot 5 x}$

Step 13.4

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

$12 \sqrt{{(3 y)}^{2} \cdot 5 x}$

Step 13.5

Add parentheses.

$12 \sqrt{{(3 y)}^{2} \cdot (5 x)}$

Step 14

Pull terms out from under the radical.

$12 (| 3 y | \sqrt{5 x})$

Step 15

Remove non-negative terms from the absolute value.

$12 (3 | y | \sqrt{5 x})$

Step 16

Multiply $3$ by $12$ .

$36 | y | \sqrt{5 x}$