Solve sqrt{2/5}:3/5(1-1/3)-sqrt{(1/2-frac{1}{3})*frac{1}{2}:3}

$$\sqrt { \frac { 2 } { 5 } } : \frac { 3 } { 5 } ( 1 - \frac { 1 } { 3 } ) - \sqrt { ( \frac { 1 } { 2 } - \frac { 1 } { 3 } ) \cdot \frac { 1 } { 2 } : 3 }$$

Evaluate

$$\frac{2\sqrt{10}}{9}-\frac{1}{6}\approx 0.536061702$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

$$\frac{\frac{\sqrt{2}}{\sqrt{5}}}{\frac{3}{5}}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\frac{3}{5}}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{\sqrt{2}\sqrt{5}}{5}}{\frac{3}{5}}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{10}}{5}}{\frac{3}{5}}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Divide \frac{\sqrt{10}}{5} by \frac{3}{5} by multiplying \frac{\sqrt{10}}{5} by the reciprocal of \frac{3}{5}.

$$\frac{\sqrt{10}\times 5}{5\times 3}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{\sqrt{10}}{3}\left(1-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\sqrt{10}}{3}\left(\frac{3}{3}-\frac{1}{3}\right)-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{10}}{3}\times \frac{3-1}{3}-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Subtract 1 from 3 to get 2.

$$\frac{\sqrt{10}}{3}\times \frac{2}{3}-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Multiply \frac{\sqrt{10}}{3} times \frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\left(\frac{1}{2}-\frac{1}{3}\right)\times \frac{1}{2}}{3}}$$

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\left(\frac{3}{6}-\frac{2}{6}\right)\times \frac{1}{2}}{3}}$$

Since \frac{3}{6} and \frac{2}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\frac{3-2}{6}\times \frac{1}{2}}{3}}$$

Subtract 2 from 3 to get 1.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\frac{1}{6}\times \frac{1}{2}}{3}}$$

Multiply \frac{1}{6} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\frac{1\times 1}{6\times 2}}{3}}$$

Do the multiplications in the fraction \frac{1\times 1}{6\times 2}.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{\frac{1}{12}}{3}}$$

Express \frac{\frac{1}{12}}{3} as a single fraction.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{1}{12\times 3}}$$

Multiply 12 and 3 to get 36.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\sqrt{\frac{1}{36}}$$

Rewrite the square root of the division \frac{1}{36} as the division of square roots \frac{\sqrt{1}}{\sqrt{36}}. Take the square root of both numerator and denominator.

$$\frac{\sqrt{10}\times 2}{3\times 3}-\frac{1}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\times 3 and 6 is 18. Multiply \frac{\sqrt{10}\times 2}{3\times 3} times \frac{2}{2}. Multiply \frac{1}{6} times \frac{3}{3}.

$$\frac{2\sqrt{10}\times 2}{18}-\frac{3}{18}$$

Since \frac{2\sqrt{10}\times 2}{18} and \frac{3}{18} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{10}\times 2-3}{18}$$

Do the multiplications in 2\sqrt{10}\times 2-3.

$$\frac{4\sqrt{10}-3}{18}$$

Factor

$$\frac{4 \sqrt{10} - 3}{18} = 0.5360617022596399$$