Simplify square root of 99-2 square root of 27+4 square root of 3-6 square root of 176

$\sqrt{99} - 2 \sqrt{27} + 4 \sqrt{3} - 6 \sqrt{176}$

Step 1

Simplify each term.

Rewrite $99$ as $3^{2} \cdot 11$ .

Factor $9$ out of $99$ .

$\sqrt{9 (11)} - 2 \sqrt{27} + 4 \sqrt{3} - 6 \sqrt{176}$

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

$\sqrt{3^{2} \cdot 11} - 2 \sqrt{27} + 4 \sqrt{3} - 6 \sqrt{176}$

Pull terms out from under the radical.

$3 \sqrt{11} - 2 \sqrt{27} + 4 \sqrt{3} - 6 \sqrt{176}$

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

Factor $9$ out of $27$ .

$3 \sqrt{11} - 2 \sqrt{9 (3)} + 4 \sqrt{3} - 6 \sqrt{176}$

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

$3 \sqrt{11} - 2 \sqrt{3^{2} \cdot 3} + 4 \sqrt{3} - 6 \sqrt{176}$

Pull terms out from under the radical.

$3 \sqrt{11} - 2 (3 \sqrt{3}) + 4 \sqrt{3} - 6 \sqrt{176}$

Multiply $3$ by $- 2$ .

$3 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3} - 6 \sqrt{176}$

Rewrite $176$ as $4^{2} \cdot 11$ .

Factor $16$ out of $176$ .

$3 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3} - 6 \sqrt{16 (11)}$

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

$3 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3} - 6 \sqrt{4^{2} \cdot 11}$

Pull terms out from under the radical.

$3 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3} - 6 (4 \sqrt{11})$

Multiply $4$ by $- 6$ .

$3 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3} - 24 \sqrt{11}$

Step 2

Simplify by adding terms.

Subtract $24 \sqrt{11}$ from $3 \sqrt{11}$ .

$- 21 \sqrt{11} - 6 \sqrt{3} + 4 \sqrt{3}$

Add $- 6 \sqrt{3}$ and $4 \sqrt{3}$ .

$- 21 \sqrt{11} - 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 21 \sqrt{11} - 2 \sqrt{3}$

Decimal Form:

$- 73.11322221 \dots$