Solve 1-sqrt{frac{2/3}}{2/3}

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

Evaluate

$$\frac{3-\sqrt{6}}{2}\approx 0.275255129$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

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

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{1-\frac{\sqrt{6}}{3}}{\frac{2}{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

$$\frac{\frac{3}{3}-\frac{\sqrt{6}}{3}}{\frac{2}{3}}$$

Since \frac{3}{3} and \frac{\sqrt{6}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3-\sqrt{6}}{3}}{\frac{2}{3}}$$

Divide \frac{3-\sqrt{6}}{3} by \frac{2}{3} by multiplying \frac{3-\sqrt{6}}{3} by the reciprocal of \frac{2}{3}.

$$\frac{\left(3-\sqrt{6}\right)\times 3}{3\times 2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-\sqrt{6}+3}{2}$$