Solve 11/5-sqrt{5/3:(71/3-2/3)^-1}+frac{3}{10}

$$\frac { 11 } { 5 } - \sqrt { \frac { 5 } { 3 } : ( \frac { 71 } { 3 } - \frac { 2 } { 3 } ) ^ { - 1 } } + \frac { 3 } { 10 }$$

Evaluate

$$-\frac{\sqrt{345}}{3}+\frac{5}{2}\approx -3.691391874$$

Short Solution Steps

Subtract \frac{2}{3} from \frac{71}{3} to get 23.

$$\frac{11}{5}-\sqrt{\frac{\frac{5}{3}}{23^{-1}}}+\frac{3}{10}$$

Calculate 23 to the power of -1 and get \frac{1}{23}.

$$\frac{11}{5}-\sqrt{\frac{\frac{5}{3}}{\frac{1}{23}}}+\frac{3}{10}$$

Divide \frac{5}{3} by \frac{1}{23} by multiplying \frac{5}{3} by the reciprocal of \frac{1}{23}.

$$\frac{11}{5}-\sqrt{\frac{5}{3}\times 23}+\frac{3}{10}$$

Multiply \frac{5}{3} and 23 to get \frac{115}{3}.

$$\frac{11}{5}-\sqrt{\frac{115}{3}}+\frac{3}{10}$$

Rewrite the square root of the division \sqrt{\frac{115}{3}} as the division of square roots \frac{\sqrt{115}}{\sqrt{3}}.

$$\frac{11}{5}-\frac{\sqrt{115}}{\sqrt{3}}+\frac{3}{10}$$

Rationalize the denominator of \frac{\sqrt{115}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{11}{5}-\frac{\sqrt{115}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{3}{10}$$

The square of \sqrt{3} is 3.

$$\frac{11}{5}-\frac{\sqrt{115}\sqrt{3}}{3}+\frac{3}{10}$$

To multiply \sqrt{115} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{11}{5}-\frac{\sqrt{345}}{3}+\frac{3}{10}$$

Add \frac{11}{5} and \frac{3}{10} to get \frac{5}{2}.

$$\frac{5}{2}-\frac{\sqrt{345}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{5}{2} times \frac{3}{3}. Multiply \frac{\sqrt{345}}{3} times \frac{2}{2}.

$$\frac{5\times 3}{6}-\frac{2\sqrt{345}}{6}$$

Since \frac{5\times 3}{6} and \frac{2\sqrt{345}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\times 3-2\sqrt{345}}{6}$$

Do the multiplications in 5\times 3-2\sqrt{345}.

$$\frac{15-2\sqrt{345}}{6}$$

Factor

$$\frac{15 - 2 \sqrt{345}}{6} = -3.6913918736689033$$