Simplify 2 square root of 5+ square root of 45+ square root of 180- square root of 80

$2 \sqrt{5} + \sqrt{45} + \sqrt{180} - \sqrt{80}$

Step 1

Simplify each term.

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

Factor $9$ out of $45$ .

$2 \sqrt{5} + \sqrt{9 (5)} + \sqrt{180} - \sqrt{80}$

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

$2 \sqrt{5} + \sqrt{3^{2} \cdot 5} + \sqrt{180} - \sqrt{80}$

Pull terms out from under the radical.

$2 \sqrt{5} + 3 \sqrt{5} + \sqrt{180} - \sqrt{80}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$2 \sqrt{5} + 3 \sqrt{5} + \sqrt{36 (5)} - \sqrt{80}$

Rewrite $36$ as $6^{2}$ .

$2 \sqrt{5} + 3 \sqrt{5} + \sqrt{6^{2} \cdot 5} - \sqrt{80}$

Pull terms out from under the radical.

$2 \sqrt{5} + 3 \sqrt{5} + 6 \sqrt{5} - \sqrt{80}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$2 \sqrt{5} + 3 \sqrt{5} + 6 \sqrt{5} - \sqrt{16 (5)}$

Rewrite $16$ as $4^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $4$ by $- 1$ .

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

Step 2

Simplify by adding terms.

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

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

Add $5 \sqrt{5}$ and $6 \sqrt{5}$ .

$11 \sqrt{5} - 4 \sqrt{5}$

Subtract $4 \sqrt{5}$ from $11 \sqrt{5}$ .

$7 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$7 \sqrt{5}$

Decimal Form:

$15.65247584 \dots$