Solve frac{sqrt{3}+1/sqrt{3}}{sqrt{3}-1/sqrt{3}}

$$\frac { \sqrt { 3 } + \frac { 1 } { \sqrt { 3 } } } { \sqrt { 3 } - \frac { 1 } { \sqrt { 3 } } }$$

Evaluate

$$2$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}+\frac{\sqrt{3}}{3}}{\sqrt{3}-\frac{1}{\sqrt{3}}}$$

Combine \sqrt{3} and \frac{\sqrt{3}}{3} to get \frac{4}{3}\sqrt{3}.

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

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

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

The square of \sqrt{3} is 3.

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

Combine \sqrt{3} and -\frac{\sqrt{3}}{3} to get \frac{2}{3}\sqrt{3}.

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

Cancel out \sqrt{3} in both numerator and denominator.

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

Divide \frac{4}{3} by \frac{2}{3} by multiplying \frac{4}{3} by the reciprocal of \frac{2}{3}.

$$\frac{4}{3}\times \frac{3}{2}$$

Multiply \frac{4}{3} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\times 3}{3\times 2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$2$$

Factor

$$2$$