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

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

Evaluate

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

Solution Steps

Rationalize the denominator of \frac{1-\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

To raise \frac{\left(1-\sqrt{3}\right)\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(\left(1-\sqrt{3}\right)\sqrt{2}\right)^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

Divide \left(4-2\sqrt{3}\right)\times 2 by 4 to get \left(4-2\sqrt{3}\right)\times \frac{1}{2}.

$$\left(4-2\sqrt{3}\right)\times \frac{1}{2}$$

Use the distributive property to multiply 4-2\sqrt{3} by \frac{1}{2}.

$$2-\sqrt{3}$$

Expand

$$2-\sqrt{3}$$

Solution Steps

Rationalize the denominator of \frac{1-\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

To raise \frac{\left(1-\sqrt{3}\right)\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(\left(1-\sqrt{3}\right)\sqrt{2}\right)^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

Divide \left(4-2\sqrt{3}\right)\times 2 by 4 to get \left(4-2\sqrt{3}\right)\times \frac{1}{2}.

$$\left(4-2\sqrt{3}\right)\times \frac{1}{2}$$

Use the distributive property to multiply 4-2\sqrt{3} by \frac{1}{2}.

$$2-\sqrt{3}$$