Solve sqrt{3/5}-frac{sqrt{5}-sqrt{6}}{sqrt{3}}

$$\sqrt { \frac { 3 } { 5 } } - \frac { \sqrt { 5 } - \sqrt { 6 } } { \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{15}\left(\sqrt{30}-2\right)}{15}\approx 0.897815783$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{5}} as the division of square roots \frac{\sqrt{3}}{\sqrt{5}}.

$$\frac{\sqrt{3}}{\sqrt{5}}-\frac{\sqrt{5}-\sqrt{6}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{\sqrt{5}-\sqrt{6}}{\sqrt{3}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}-\sqrt{6}}{\sqrt{3}}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{15}}{5}-\frac{\sqrt{5}-\sqrt{6}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{5}-\sqrt{6}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{15}}{5}-\frac{\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{15}}{5}-\frac{\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{\sqrt{15}}{5} times \frac{3}{3}. Multiply \frac{\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{3} times \frac{5}{5}.

$$\frac{3\sqrt{15}}{15}-\frac{5\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{15}$$

Since \frac{3\sqrt{15}}{15} and \frac{5\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\sqrt{15}-5\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}}{15}$$

Do the multiplications in 3\sqrt{15}-5\left(\sqrt{5}-\sqrt{6}\right)\sqrt{3}.

$$\frac{3\sqrt{15}-5\sqrt{15}+15\sqrt{2}}{15}$$

Do the calculations in 3\sqrt{15}-5\sqrt{15}+15\sqrt{2}.

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