Simplify square root of 3y^2+4 square root of 12y^2-y square root of 75

$\sqrt{3 y^{2}} + 4 \sqrt{12 y^{2}} - y \sqrt{75}$

Step 1

Simplify each term.

Reorder $3$ and $y^{2}$ .

$\sqrt{y^{2} \cdot 3} + 4 \sqrt{12 y^{2}} - y \sqrt{75}$

Pull terms out from under the radical.

$y \sqrt{3} + 4 \sqrt{12 y^{2}} - y \sqrt{75}$

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

Factor $4$ out of $12$ .

$y \sqrt{3} + 4 \sqrt{4 (3) y^{2}} - y \sqrt{75}$

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

$y \sqrt{3} + 4 \sqrt{2^{2} \cdot 3 y^{2}} - y \sqrt{75}$

Move $3$ .

$y \sqrt{3} + 4 \sqrt{2^{2} y^{2} \cdot 3} - y \sqrt{75}$

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

$y \sqrt{3} + 4 \sqrt{{(2 y)}^{2} \cdot 3} - y \sqrt{75}$

Pull terms out from under the radical.

$y \sqrt{3} + 4 (2 y \sqrt{3}) - y \sqrt{75}$

Multiply $2$ by $4$ .

$y \sqrt{3} + 8 (y \sqrt{3}) - y \sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$y \sqrt{3} + 8 y \sqrt{3} - y \sqrt{25 (3)}$

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

$y \sqrt{3} + 8 y \sqrt{3} - y \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$y \sqrt{3} + 8 y \sqrt{3} - y (5 \sqrt{3})$

Multiply $5$ by $- 1$ .

$y \sqrt{3} + 8 y \sqrt{3} - 5 y \sqrt{3}$

Step 2

Simplify by adding terms.

Add $y \sqrt{3}$ and $8 y \sqrt{3}$ .

$9 y \sqrt{3} - 5 y \sqrt{3}$

Subtract $5 y \sqrt{3}$ from $9 y \sqrt{3}$ .

$4 y \sqrt{3}$