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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

Since \frac{3\times 2\sqrt{5}}{3} and \frac{-2\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\times 2\sqrt{5}-2\sqrt{3}.

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{5}}-\sqrt{75}\right)$$

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

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{\sqrt{1}}{\sqrt{5}}-\sqrt{75}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{1}{\sqrt{5}}-\sqrt{75}\right)$$

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

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\sqrt{75}\right)$$

The square of \sqrt{5} is 5.

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{\sqrt{5}}{5}-\sqrt{75}\right)$$

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}.

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{\sqrt{5}}{5}-5\sqrt{3}\right)$$

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

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\left(\frac{\sqrt{5}}{5}+\frac{5\left(-5\right)\sqrt{3}}{5}\right)$$

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

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\frac{\sqrt{5}+5\left(-5\right)\sqrt{3}}{5}$$

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

$$\frac{6\sqrt{5}-2\sqrt{3}}{3}-\frac{\sqrt{5}-25\sqrt{3}}{5}$$

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}-2\sqrt{3}}{3} times \frac{5}{5}. Multiply \frac{\sqrt{5}-25\sqrt{3}}{5} times \frac{3}{3}.

$$\frac{5\left(6\sqrt{5}-2\sqrt{3}\right)}{15}-\frac{3\left(\sqrt{5}-25\sqrt{3}\right)}{15}$$

Since \frac{5\left(6\sqrt{5}-2\sqrt{3}\right)}{15} and \frac{3\left(\sqrt{5}-25\sqrt{3}\right)}{15} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(6\sqrt{5}-2\sqrt{3}\right)-3\left(\sqrt{5}-25\sqrt{3}\right)}{15}$$

Do the multiplications in 5\left(6\sqrt{5}-2\sqrt{3}\right)-3\left(\sqrt{5}-25\sqrt{3}\right).

$$\frac{30\sqrt{5}-10\sqrt{3}-3\sqrt{5}+75\sqrt{3}}{15}$$

Do the calculations in 30\sqrt{5}-10\sqrt{3}-3\sqrt{5}+75\sqrt{3}.

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