Solve =sqrt{2/5}+3sqrt{5}-4sqrt{5}

$$= \sqrt { \frac { 2 } { 5 } } + 3 \sqrt { 5 } - 4 \sqrt { 5 }$$

Evaluate

$$\frac{\sqrt{10}}{5}-\sqrt{5}\approx -1.603612445$$

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}}.

$$\frac{\sqrt{2}}{\sqrt{5}}+3\sqrt{5}-4\sqrt{5}$$

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

$$\frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+3\sqrt{5}-4\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{2}\sqrt{5}}{5}+3\sqrt{5}-4\sqrt{5}$$

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

$$\frac{\sqrt{10}}{5}+3\sqrt{5}-4\sqrt{5}$$

Combine 3\sqrt{5} and -4\sqrt{5} to get -\sqrt{5}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{5} times \frac{5}{5}.

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

Since \frac{\sqrt{10}}{5} and \frac{5\sqrt{5}}{5} have the same denominator, subtract them by subtracting their numerators.

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