Simplify the Radical Expression square root of 500+ square root of 20+11 square root of 5

$\sqrt{500} + \sqrt{20} + 11 \sqrt{5}$

Simplify each term.

Rewrite $500$ as $10^{2} \cdot 5$ .

Factor $100$ out of $500$ .

$\sqrt{100 (5)} + \sqrt{20} + 11 \sqrt{5}$

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2} \cdot 5} + \sqrt{20} + 11 \sqrt{5}$

Pull terms out from under the radical.

$| 10 | \sqrt{5} + \sqrt{20} + 11 \sqrt{5}$

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$10 \sqrt{5} + \sqrt{20} + 11 \sqrt{5}$

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

Factor $4$ out of $20$ .

$10 \sqrt{5} + \sqrt{4 (5)} + 11 \sqrt{5}$

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

$10 \sqrt{5} + \sqrt{2^{2} \cdot 5} + 11 \sqrt{5}$

Pull terms out from under the radical.

$10 \sqrt{5} + | 2 | \sqrt{5} + 11 \sqrt{5}$

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

$10 \sqrt{5} + 2 \sqrt{5} + 11 \sqrt{5}$

Add $10 \sqrt{5}$ and $2 \sqrt{5}$ .

$12 \sqrt{5} + 11 \sqrt{5}$

Add $12 \sqrt{5}$ and $11 \sqrt{5}$ .

$23 \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$23 \sqrt{5}$

Decimal Form:

$51.42956348 \dots$