Solve (1/sqrt{5}+sqrt{20}-3sqrt{5})*sqrt{10}

$$( \frac { 1 } { \sqrt { 5 } } + \sqrt { 20 } - 3 \sqrt { 5 } ) \times \sqrt { 10 }$$

Evaluate

$$-4\sqrt{2}\approx -5.656854249$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\left(\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\sqrt{20}-3\sqrt{5}\right)\sqrt{10}$$

The square of \sqrt{5} is 5.

$$\left(\frac{\sqrt{5}}{5}+\sqrt{20}-3\sqrt{5}\right)\sqrt{10}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

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

Combine \frac{\sqrt{5}}{5} and 2\sqrt{5} to get \frac{11}{5}\sqrt{5}.

$$\left(\frac{11}{5}\sqrt{5}-3\sqrt{5}\right)\sqrt{10}$$

Combine \frac{11}{5}\sqrt{5} and -3\sqrt{5} to get -\frac{4}{5}\sqrt{5}.

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

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$-\frac{4}{5}\times 5\sqrt{2}$$

Cancel out 5 and 5.

$$-4\sqrt{2}$$