Solve 4/(sqrt{3)^2}+1/(sqrt{3/2)^2}-(1/sqrt{2})^2

$$\frac { 4 } { ( \sqrt { 3 } ) ^ { 2 } } + \frac { 1 } { ( \frac { \sqrt { 3 } } { 2 } ) ^ { 2 } } - ( \frac { 1 } { \sqrt { 2 } } ) ^ { 2 }$$

Evaluate

$$\frac{13}{6}\approx 2.166666667$$

Solution Steps

The square of \sqrt{3} is 3.

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

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Divide 1 by \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} by multiplying 1 by the reciprocal of \frac{\left(\sqrt{3}\right)^{2}}{2^{2}}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Add \frac{4}{3} and \frac{4}{3} to get \frac{8}{3}.

$$\frac{8}{3}-\left(\frac{1}{\sqrt{2}}\right)^{2}$$

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

$$\frac{8}{3}-\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{8}{3}-\left(\frac{\sqrt{2}}{2}\right)^{2}$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8}{3}-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8}{3}-\frac{2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8}{3}-\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{8}{3}-\frac{1}{2}$$

Subtract \frac{1}{2} from \frac{8}{3} to get \frac{13}{6}.

$$\frac{13}{6}$$

Factor

$$\frac{13}{2 \cdot 3} = 2\frac{1}{6} = 2.1666666666666665$$