Solve 3sqrt{8/3-1/2}-frac{sqrt{2}}{15}

$$3 \sqrt { \frac { 8 } { 3 } - \frac { 1 } { 2 } } - \frac { \sqrt { 2 } } { 15 }$$

Evaluate

$$\frac{\sqrt{78}}{2}-\frac{\sqrt{2}}{15}\approx 4.321599529$$

Short Solution Steps

Least common multiple of 3 and 2 is 6. Convert \frac{8}{3} and \frac{1}{2} to fractions with denominator 6.

$$3\sqrt{\frac{16}{6}-\frac{3}{6}}-\frac{\sqrt{2}}{15}$$

Since \frac{16}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

$$3\sqrt{\frac{16-3}{6}}-\frac{\sqrt{2}}{15}$$

Subtract 3 from 16 to get 13.

$$3\sqrt{\frac{13}{6}}-\frac{\sqrt{2}}{15}$$

Rewrite the square root of the division \sqrt{\frac{13}{6}} as the division of square roots \frac{\sqrt{13}}{\sqrt{6}}.

$$3\times \frac{\sqrt{13}}{\sqrt{6}}-\frac{\sqrt{2}}{15}$$

Rationalize the denominator of \frac{\sqrt{13}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$3\times \frac{\sqrt{13}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}-\frac{\sqrt{2}}{15}$$

The square of \sqrt{6} is 6.

$$3\times \frac{\sqrt{13}\sqrt{6}}{6}-\frac{\sqrt{2}}{15}$$

To multiply \sqrt{13} and \sqrt{6}, multiply the numbers under the square root.

$$3\times \frac{\sqrt{78}}{6}-\frac{\sqrt{2}}{15}$$

Cancel out 6, the greatest common factor in 3 and 6.

$$\frac{\sqrt{78}}{2}-\frac{\sqrt{2}}{15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 15 is 30. Multiply \frac{\sqrt{78}}{2} times \frac{15}{15}. Multiply \frac{\sqrt{2}}{15} times \frac{2}{2}.

$$\frac{15\sqrt{78}}{30}-\frac{2\sqrt{2}}{30}$$

Since \frac{15\sqrt{78}}{30} and \frac{2\sqrt{2}}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15\sqrt{78}-2\sqrt{2}}{30}$$

Factor

$$\frac{15 \sqrt{78} - 2 \sqrt{2}}{30} = 4.321599529005718$$