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

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

Step 1

Simplify each term.

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

Factor $9$ out of $45$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $3$ by $- 2$ .

$- 6 \sqrt{5} - 3 \sqrt{20} - 2 \sqrt{6}$

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

Factor $4$ out of $20$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $- 3$ .

$- 6 \sqrt{5} - 6 \sqrt{5} - 2 \sqrt{6}$

Step 2

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

$- 12 \sqrt{5} - 2 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 12 \sqrt{5} - 2 \sqrt{6}$

Decimal Form:

$- 31.73179521 \dots$