Solve frac{sqrt{3}}{sqrt{5}}-frac{sqrt{5}+sqrt{30}}{sqrt{2}}

$$\frac { \sqrt { 3 } } { \sqrt { 5 } } - \frac { \sqrt { 5 } + \sqrt { 30 } } { \sqrt { 2 } }$$

Evaluate

$$-\frac{\sqrt{10}}{2}-\frac{4\sqrt{15}}{5}\approx -4.679525507$$

Short Solution Steps

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

$$\frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{\sqrt{5}+\sqrt{30}}{\sqrt{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}-\frac{\sqrt{5}+\sqrt{30}}{\sqrt{2}}$$

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

$$\frac{\sqrt{15}}{5}-\frac{\sqrt{5}+\sqrt{30}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{5}+\sqrt{30}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{15}}{5}-\frac{\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{15}}{5}-\frac{\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 2 is 10. Multiply \frac{\sqrt{15}}{5} times \frac{2}{2}. Multiply \frac{\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{2} times \frac{5}{5}.

$$\frac{2\sqrt{15}}{10}-\frac{5\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{10}$$

Since \frac{2\sqrt{15}}{10} and \frac{5\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{15}-5\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}}{10}$$

Do the multiplications in 2\sqrt{15}-5\left(\sqrt{5}+\sqrt{30}\right)\sqrt{2}.

$$\frac{2\sqrt{15}-5\sqrt{10}-10\sqrt{15}}{10}$$

Do the calculations in 2\sqrt{15}-5\sqrt{10}-10\sqrt{15}.

$$\frac{-8\sqrt{15}-5\sqrt{10}}{10}$$