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

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

Evaluate

$$2-\sqrt{3}\approx 0.267949192$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

$$\frac{4-2\sqrt{3}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4-2\sqrt{3}}{2}$$

Divide each term of 4-2\sqrt{3} by 2 to get 2-\sqrt{3}.

$$2-\sqrt{3}$$