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

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

Evaluate

$$5\sqrt{5}-15\approx -3.819660113$$

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{5} is 5.

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

Cancel out 5 and 5.

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

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

$$5\sqrt{5}-3\sqrt{\frac{1}{5}}\sqrt{125}$$

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{5} is 5.

$$5\sqrt{5}-3\times \frac{\sqrt{5}}{5}\sqrt{125}$$

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}.

$$5\sqrt{5}-3\times \frac{\sqrt{5}}{5}\times 5\sqrt{5}$$

Multiply 3 and 5 to get 15.

$$5\sqrt{5}-15\times \frac{\sqrt{5}}{5}\sqrt{5}$$

Cancel out 5, the greatest common factor in 15 and 5.

$$5\sqrt{5}-3\sqrt{5}\sqrt{5}$$

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

$$5\sqrt{5}-3\times 5$$

Multiply 3 and 5 to get 15.

$$5\sqrt{5}-15$$