Solve sqrt{1/5}+1/2sqrt{20}+5/4sqrt{4/5}+sqrt{45}

$$\sqrt { \frac { 1 } { 5 } } + \frac { 1 } { 2 } \sqrt { 20 } + \frac { 5 } { 4 } \sqrt { \frac { 4 } { 5 } } + \sqrt { 45 }$$

Evaluate

$$\frac{47\sqrt{5}}{10}\approx 10.509519494$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$\frac{\sqrt{1}}{\sqrt{5}}+\frac{1}{2}\sqrt{20}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{5}}+\frac{1}{2}\sqrt{20}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

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

$$\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\frac{1}{2}\sqrt{20}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}}{5}+\frac{1}{2}\sqrt{20}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

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}.

$$\frac{\sqrt{5}}{5}+\frac{1}{2}\times 2\sqrt{5}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

Cancel out 2 and 2.

$$\frac{\sqrt{5}}{5}+\sqrt{5}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

Combine \frac{\sqrt{5}}{5} and \sqrt{5} to get \frac{6}{5}\sqrt{5}.

$$\frac{6}{5}\sqrt{5}+\frac{5}{4}\sqrt{\frac{4}{5}}+\sqrt{45}$$

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

$$\frac{6}{5}\sqrt{5}+\frac{5}{4}\times \frac{\sqrt{4}}{\sqrt{5}}+\sqrt{45}$$

Calculate the square root of 4 and get 2.

$$\frac{6}{5}\sqrt{5}+\frac{5}{4}\times \frac{2}{\sqrt{5}}+\sqrt{45}$$

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

$$\frac{6}{5}\sqrt{5}+\frac{5}{4}\times \frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\sqrt{45}$$

The square of \sqrt{5} is 5.

$$\frac{6}{5}\sqrt{5}+\frac{5}{4}\times \frac{2\sqrt{5}}{5}+\sqrt{45}$$

Multiply \frac{5}{4} times \frac{2\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{6}{5}\sqrt{5}+\frac{5\times 2\sqrt{5}}{4\times 5}+\sqrt{45}$$

Cancel out 2\times 5 in both numerator and denominator.

$$\frac{6}{5}\sqrt{5}+\frac{\sqrt{5}}{2}+\sqrt{45}$$

Combine \frac{6}{5}\sqrt{5} and \frac{\sqrt{5}}{2} to get \frac{17}{10}\sqrt{5}.

$$\frac{17}{10}\sqrt{5}+\sqrt{45}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{17}{10}\sqrt{5}+3\sqrt{5}$$

Combine \frac{17}{10}\sqrt{5} and 3\sqrt{5} to get \frac{47}{10}\sqrt{5}.

$$\frac{47}{10}\sqrt{5}$$