Solve sqrt{20}-2sqrt{1/3}-sqrt{1/5}+sqrt{75}

$$\sqrt{ 20 } -2 \sqrt{ \frac{ 1 }{ 3 } } - \sqrt{ \frac{ 1 }{ 5 } } + \sqrt{ 75 }$$

Evaluate

$$\frac{9\sqrt{5}}{5}+\frac{13\sqrt{3}}{3}\approx 11.530475859$$

Short 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}.

$$2\sqrt{5}-2\sqrt{\frac{1}{3}}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$2\sqrt{5}-2\times \frac{\sqrt{1}}{\sqrt{3}}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{5}-2\times \frac{1}{\sqrt{3}}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$2\sqrt{5}-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

The square of \sqrt{3} is 3.

$$2\sqrt{5}-2\times \frac{\sqrt{3}}{3}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

Express -2\times \frac{\sqrt{3}}{3} as a single fraction.

$$2\sqrt{5}+\frac{-2\sqrt{3}}{3}-\sqrt{\frac{1}{5}}+\sqrt{75}$$

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

$$2\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{1}}{\sqrt{5}}+\sqrt{75}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{1}{\sqrt{5}}+\sqrt{75}$$

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

$$2\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\sqrt{75}$$

The square of \sqrt{5} is 5.

$$2\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{5}}{5}+\sqrt{75}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{5}+5\sqrt{3} times \frac{3}{3}.

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

Since \frac{3\left(2\sqrt{5}+5\sqrt{3}\right)}{3} and \frac{-2\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(2\sqrt{5}+5\sqrt{3}\right)-2\sqrt{3}.

$$\frac{6\sqrt{5}+15\sqrt{3}-2\sqrt{3}}{3}-\frac{\sqrt{5}}{5}$$

Do the calculations in 6\sqrt{5}+15\sqrt{3}-2\sqrt{3}.

$$\frac{6\sqrt{5}+13\sqrt{3}}{3}-\frac{\sqrt{5}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5 is 15. Multiply \frac{6\sqrt{5}+13\sqrt{3}}{3} times \frac{5}{5}. Multiply \frac{\sqrt{5}}{5} times \frac{3}{3}.

$$\frac{5\left(6\sqrt{5}+13\sqrt{3}\right)}{15}-\frac{3\sqrt{5}}{15}$$

Since \frac{5\left(6\sqrt{5}+13\sqrt{3}\right)}{15} and \frac{3\sqrt{5}}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(6\sqrt{5}+13\sqrt{3}\right)-3\sqrt{5}}{15}$$

Do the multiplications in 5\left(6\sqrt{5}+13\sqrt{3}\right)-3\sqrt{5}.

$$\frac{30\sqrt{5}+65\sqrt{3}-3\sqrt{5}}{15}$$

Do the calculations in 30\sqrt{5}+65\sqrt{3}-3\sqrt{5}.

$$\frac{27\sqrt{5}+65\sqrt{3}}{15}$$