Solve 3sqrt{20}-sqrt{45}+10sqrt{1/5}

$$3 \sqrt { 20 } - \sqrt { 45 } + 10 \sqrt { \frac { 1 } { 5 } }$$

Evaluate

$$5\sqrt{5}\approx 11.180339887$$

Solution Steps

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}.

$$3\times 2\sqrt{5}-\sqrt{45}+10\sqrt{\frac{1}{5}}$$

Multiply 3 and 2 to get 6.

$$6\sqrt{5}-\sqrt{45}+10\sqrt{\frac{1}{5}}$$

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}.

$$6\sqrt{5}-3\sqrt{5}+10\sqrt{\frac{1}{5}}$$

Combine 6\sqrt{5} and -3\sqrt{5} to get 3\sqrt{5}.

$$3\sqrt{5}+10\sqrt{\frac{1}{5}}$$

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

$$3\sqrt{5}+10\times \frac{\sqrt{1}}{\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{5}+10\times \frac{1}{\sqrt{5}}$$

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

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

The square of \sqrt{5} is 5.

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

Cancel out 5, the greatest common factor in 10 and 5.

$$3\sqrt{5}+2\sqrt{5}$$

Combine 3\sqrt{5} and 2\sqrt{5} to get 5\sqrt{5}.

$$5\sqrt{5}$$