Simplify the Radical Expression -3 square root of 45+2 square root of 12+3 square root of 6-3 square root of 20

$- 3 \sqrt{45} + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

Simplify each term.

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

Factor $9$ out of $45$ .

$- 3 \sqrt{9 (5)} + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

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

$- 3 \sqrt{3^{2} \cdot 5} + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

Pull terms out from under the radical.

$- 3 (| 3 | \sqrt{5}) + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

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

$- 3 (3 \sqrt{5}) + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

Multiply $3$ by $- 3$ .

$- 9 \sqrt{5} + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

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

Factor $4$ out of $12$ .

$- 9 \sqrt{5} + 2 \sqrt{4 (3)} + 3 \sqrt{6} - 3 \sqrt{20}$

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

$- 9 \sqrt{5} + 2 \sqrt{2^{2} \cdot 3} + 3 \sqrt{6} - 3 \sqrt{20}$

Pull terms out from under the radical.

$- 9 \sqrt{5} + 2 (| 2 | \sqrt{3}) + 3 \sqrt{6} - 3 \sqrt{20}$

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

$- 9 \sqrt{5} + 2 (2 \sqrt{3}) + 3 \sqrt{6} - 3 \sqrt{20}$

Multiply $2$ by $2$ .

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 3 \sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 3 \sqrt{4 (5)}$

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

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 3 \sqrt{2^{2} \cdot 5}$

Pull terms out from under the radical.

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 3 (| 2 | \sqrt{5})$

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

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 3 (2 \sqrt{5})$

Multiply $2$ by $- 3$ .

$- 9 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6} - 6 \sqrt{5}$

Subtract $6 \sqrt{5}$ from $- 9 \sqrt{5}$ .

$- 15 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6}$

The result can be shown in multiple forms.

Exact Form:

$- 15 \sqrt{5} + 4 \sqrt{3} + 3 \sqrt{6}$

Decimal Form:

$- 19.26434720 \dots$