Evaluate - square root of 45+2 square root of 5- square root of 20-2 square root of 6

$- \sqrt{45} + 2 \sqrt{5} - \sqrt{20} - 2 \sqrt{6}$

Step 1

Simplify each term.

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

Factor $9$ out of $45$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $3$ by $- 1$ .

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

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

Factor $4$ out of $20$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Step 2

Simplify by adding terms.

Add $- 3 \sqrt{5}$ and $2 \sqrt{5}$ .

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

Subtract $2 \sqrt{5}$ from $- \sqrt{5}$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 11.60718341 \dots$