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

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

Step 1

Simplify each term.

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

Factor $4$ out of $20$ .

$2 \sqrt{4 (5)} - \sqrt{20} + 3 \sqrt{20} - 2 \sqrt{45}$

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

$2 \sqrt{2^{2} \cdot 5} - \sqrt{20} + 3 \sqrt{20} - 2 \sqrt{45}$

Pull terms out from under the radical.

$2 (| 2 | \sqrt{5}) - \sqrt{20} + 3 \sqrt{20} - 2 \sqrt{45}$

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

$2 (2 \sqrt{5}) - \sqrt{20} + 3 \sqrt{20} - 2 \sqrt{45}$

Multiply $2$ by $2$ .

$4 \sqrt{5} - \sqrt{20} + 3 \sqrt{20} - 2 \sqrt{45}$

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

Factor $4$ out of $20$ .

$4 \sqrt{5} - \sqrt{4 (5)} + 3 \sqrt{20} - 2 \sqrt{45}$

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

$4 \sqrt{5} - \sqrt{2^{2} \cdot 5} + 3 \sqrt{20} - 2 \sqrt{45}$

Pull terms out from under the radical.

$4 \sqrt{5} - (| 2 | \sqrt{5}) + 3 \sqrt{20} - 2 \sqrt{45}$

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

$4 \sqrt{5} - (2 \sqrt{5}) + 3 \sqrt{20} - 2 \sqrt{45}$

Multiply $2$ by $- 1$ .

$4 \sqrt{5} - 2 \sqrt{5} + 3 \sqrt{20} - 2 \sqrt{45}$

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

Factor $4$ out of $20$ .

$4 \sqrt{5} - 2 \sqrt{5} + 3 \sqrt{4 (5)} - 2 \sqrt{45}$

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

$4 \sqrt{5} - 2 \sqrt{5} + 3 \sqrt{2^{2} \cdot 5} - 2 \sqrt{45}$

Pull terms out from under the radical.

$4 \sqrt{5} - 2 \sqrt{5} + 3 (| 2 | \sqrt{5}) - 2 \sqrt{45}$

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

$4 \sqrt{5} - 2 \sqrt{5} + 3 (2 \sqrt{5}) - 2 \sqrt{45}$

Multiply $2$ by $3$ .

$4 \sqrt{5} - 2 \sqrt{5} + 6 \sqrt{5} - 2 \sqrt{45}$

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

Factor $9$ out of $45$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $3$ by $- 2$ .

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

Step 2

Subtract $2 \sqrt{5}$ from $4 \sqrt{5}$ .

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

Step 3

Add $2 \sqrt{5}$ and $6 \sqrt{5}$ .

$8 \sqrt{5} - 6 \sqrt{5}$

Step 4

Subtract $6 \sqrt{5}$ from $8 \sqrt{5}$ .

$2 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{5}$

Decimal Form:

$4.47213595 \dots$

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

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