Basic Math Examples

$\sqrt{24 y^{3} + 12 y^{2}} + \sqrt{15 (5 + 10 y)}$

Step 1

Factor $12 y^{2}$ out of $24 y^{3} + 12 y^{2}$ .

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Step 1.1

Factor $12 y^{2}$ out of $24 y^{3}$ .

$\sqrt{12 y^{2} (2 y) + 12 y^{2}} + \sqrt{15 (5 + 10 y)}$

Step 1.2

Factor $12 y^{2}$ out of $12 y^{2}$ .

$\sqrt{12 y^{2} (2 y) + 12 y^{2} (1)} + \sqrt{15 (5 + 10 y)}$

Step 1.3

Factor $12 y^{2}$ out of $12 y^{2} (2 y) + 12 y^{2} (1)$ .

$\sqrt{12 y^{2} (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 2

Rewrite $12 y^{2} (2 y + 1)$ as ${(2 y)}^{2} \cdot (3 (2 y + 1))$ .

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Step 2.1

Factor $4$ out of $12$ .

$\sqrt{4 (3) y^{2} (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 2.2

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

$\sqrt{2^{2} \cdot 3 y^{2} (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 2.3

Move $3$ .

$\sqrt{2^{2} y^{2} \cdot 3 (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 2.4

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

$\sqrt{{(2 y)}^{2} \cdot 3 (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 2.5

Add parentheses.

$\sqrt{{(2 y)}^{2} \cdot (3 (2 y + 1))} + \sqrt{15 (5 + 10 y)}$

Step 3

Pull terms out from under the radical.

$2 y \sqrt{3 (2 y + 1)} + \sqrt{15 (5 + 10 y)}$

Step 4

Factor $5$ out of $5 + 10 y$ .

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Step 4.1

Factor $5$ out of $5$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{15 (5 (1) + 10 y)}$

Step 4.2

Factor $5$ out of $10 y$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{15 (5 (1) + 5 (2 y))}$

Step 4.3

Factor $5$ out of $5 (1) + 5 (2 y)$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{15 (5 (1 + 2 y))}$

$2 y \sqrt{3 (2 y + 1)} + \sqrt{15 \cdot 5 (1 + 2 y)}$

Step 5

Multiply $15$ by $5$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{75 (1 + 2 y)}$

Step 6

Rewrite $75 (1 + 2 y)$ as $5^{2} \cdot (3 (1^{2} + 2 y))$ .

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Step 6.1

Factor $25$ out of $75$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{25 (3) (1 + 2 y)}$

Step 6.2

Rewrite $25$ as $5^{2}$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{5^{2} \cdot 3 (1 + 2 y)}$

Step 6.3

Rewrite $1$ as $1^{2}$ .

$2 y \sqrt{3 (2 y + 1)} + \sqrt{5^{2} \cdot 3 (1^{2} + 2 y)}$

Step 6.4

Add parentheses.

$2 y \sqrt{3 (2 y + 1)} + \sqrt{5^{2} \cdot (3 (1^{2} + 2 y))}$

Step 7

Pull terms out from under the radical.

$2 y \sqrt{3 (2 y + 1)} + 5 \sqrt{3 (1^{2} + 2 y)}$

Step 8

One to any power is one.

$2 y \sqrt{3 (2 y + 1)} + 5 \sqrt{3 (1 + 2 y)}$