Solve {left(1-sqrt{3}right)}^2+2-sqrt{3}-1+sqrt{108}

$${ \left(1- \sqrt{ 3 } \right) }^{ 2 } +2- \sqrt{ 3 } -1+ \sqrt{ 108 }$$

Evaluate

$$3\sqrt{3}+5\approx 10.196152423$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{3}\right)^{2}.

$$1-2\sqrt{3}+\left(\sqrt{3}\right)^{2}+2-\sqrt{3}-1+\sqrt{108}$$

The square of \sqrt{3} is 3.

$$1-2\sqrt{3}+3+2-\sqrt{3}-1+\sqrt{108}$$

Add 1 and 3 to get 4.

$$4-2\sqrt{3}+2-\sqrt{3}-1+\sqrt{108}$$

Add 4 and 2 to get 6.

$$6-2\sqrt{3}-\sqrt{3}-1+\sqrt{108}$$

Combine -2\sqrt{3} and -\sqrt{3} to get -3\sqrt{3}.

$$6-3\sqrt{3}-1+\sqrt{108}$$

Subtract 1 from 6 to get 5.

$$5-3\sqrt{3}+\sqrt{108}$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

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

Combine -3\sqrt{3} and 6\sqrt{3} to get 3\sqrt{3}.

$$5+3\sqrt{3}$$