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

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

Evaluate

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Solution Steps

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

The square of \sqrt{3} is 3.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}-\frac{3}{9}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{1}{3} times \frac{3}{3}.

$$\frac{\left(2\sqrt{3}\right)^{2}}{9}-\frac{3}{9}$$

Since \frac{\left(2\sqrt{3}\right)^{2}}{9} and \frac{3}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2\sqrt{3}\right)^{2}-3}{9}$$

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{12}{9}-\frac{1}{3}$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

$$\frac{4}{3}-\frac{1}{3}$$

Subtract \frac{1}{3} from \frac{4}{3} to get 1.

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Factor

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