Solve sqrt{80}+5sqrt{1/2}-3sqrt{5}+1/5sqrt{125}

$$\sqrt{ 80 } +5 \sqrt{ \frac{ 1 }{ 2 } } -3 \sqrt{ 5 } + \frac{ 1 }{ 5 } \sqrt{ 125 }$$

Evaluate

$$\frac{\sqrt{5}\left(\sqrt{10}+4\right)}{2}\approx 8.007669861$$

Short Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$4\sqrt{5}+5\sqrt{\frac{1}{2}}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$4\sqrt{5}+5\times \frac{\sqrt{1}}{\sqrt{2}}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{5}+5\times \frac{1}{\sqrt{2}}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{5}+5\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

The square of \sqrt{2} is 2.

$$4\sqrt{5}+5\times \frac{\sqrt{2}}{2}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

Express 5\times \frac{\sqrt{2}}{2} as a single fraction.

$$4\sqrt{5}+\frac{5\sqrt{2}}{2}-3\sqrt{5}+\frac{1}{5}\sqrt{125}$$

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

$$\sqrt{5}+\frac{5\sqrt{2}}{2}+\frac{1}{5}\sqrt{125}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

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

Cancel out 5 and 5.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{5} times \frac{2}{2}.

$$\frac{2\times 2\sqrt{5}}{2}+\frac{5\sqrt{2}}{2}$$

Since \frac{2\times 2\sqrt{5}}{2} and \frac{5\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2\sqrt{5}+5\sqrt{2}}{2}$$

Do the multiplications in 2\times 2\sqrt{5}+5\sqrt{2}.

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

Factor

$$\frac{\sqrt{5} {(\sqrt{2} \sqrt{5} + 4)}}{2} = 8.007669860932317$$