Simplify the Radical Expression - square root of 27-3 square root of 45- square root of 20+2 square root of 45

$- \sqrt{27} - 3 \sqrt{45} - \sqrt{20} + 2 \sqrt{45}$

Step 1

Simplify each term.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$- \sqrt{9 (3)} - 3 \sqrt{45} - \sqrt{20} + 2 \sqrt{45}$

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

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

Pull terms out from under the radical.

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

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

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

Multiply $3$ by $- 1$ .

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

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

Factor $9$ out of $45$ .

$- 3 \sqrt{3} - 3 \sqrt{9 (5)} - \sqrt{20} + 2 \sqrt{45}$

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

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

Pull terms out from under the radical.

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

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

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

Multiply $3$ by $- 3$ .

$- 3 \sqrt{3} - 9 \sqrt{5} - \sqrt{20} + 2 \sqrt{45}$

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

Factor $4$ out of $20$ .

$- 3 \sqrt{3} - 9 \sqrt{5} - \sqrt{4 (5)} + 2 \sqrt{45}$

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

$- 3 \sqrt{3} - 9 \sqrt{5} - \sqrt{2^{2} \cdot 5} + 2 \sqrt{45}$

Pull terms out from under the radical.

$- 3 \sqrt{3} - 9 \sqrt{5} - (| 2 | \sqrt{5}) + 2 \sqrt{45}$

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

$- 3 \sqrt{3} - 9 \sqrt{5} - (2 \sqrt{5}) + 2 \sqrt{45}$

Multiply $2$ by $- 1$ .

$- 3 \sqrt{3} - 9 \sqrt{5} - 2 \sqrt{5} + 2 \sqrt{45}$

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

Factor $9$ out of $45$ .

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

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

$- 3 \sqrt{3} - 9 \sqrt{5} - 2 \sqrt{5} + 2 \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$- 3 \sqrt{3} - 9 \sqrt{5} - 2 \sqrt{5} + 2 (| 3 | \sqrt{5})$

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

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

Multiply $3$ by $2$ .

$- 3 \sqrt{3} - 9 \sqrt{5} - 2 \sqrt{5} + 6 \sqrt{5}$

Step 2

Subtract $2 \sqrt{5}$ from $- 9 \sqrt{5}$ .

$- 3 \sqrt{3} - 11 \sqrt{5} + 6 \sqrt{5}$

Step 3

Add $- 11 \sqrt{5}$ and $6 \sqrt{5}$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 16.37649231 \dots$

Simplify the Radical Expression - square root of 27-3 square root of 45- square root of 20+2 square root of 45

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