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

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

Evaluate

$$-\frac{13}{3}\approx -4.333333333$$

Solution Steps

The square of \sqrt{3} is 3.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Multiply 2 and \frac{1}{4} to get \frac{1}{2}.

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

Subtract \frac{1}{2} from 3 to get \frac{5}{2}.

$$\frac{5}{2}-\frac{3}{4}\left(\sqrt{2}\right)^{2}-4\times \left(\frac{2}{\sqrt{3}}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{5}{2}-\frac{3}{4}\times 2-4\times \left(\frac{2}{\sqrt{3}}\right)^{2}$$

Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

$$\frac{5}{2}-\frac{3}{2}-4\times \left(\frac{2}{\sqrt{3}}\right)^{2}$$

Subtract \frac{3}{2} from \frac{5}{2} to get 1.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Express 4\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}} as a single fraction.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$1-\frac{4\times 4\times 3}{3^{2}}$$

Multiply 4 and 3 to get 12.

$$1-\frac{4\times 12}{3^{2}}$$

Multiply 4 and 12 to get 48.

$$1-\frac{48}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$1-\frac{48}{9}$$

Reduce the fraction \frac{48}{9} to lowest terms by extracting and canceling out 3.

$$1-\frac{16}{3}$$

Subtract \frac{16}{3} from 1 to get -\frac{13}{3}.

$$-\frac{13}{3}$$

Factor

$$-\frac{13}{3} = -4\frac{1}{3} = -4.333333333333333$$