Solve frac{sqrt{75}-sqrt{3}}{sqrt{3}}-sqrt{1/5}*sqrt{20}

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

Evaluate

$$2$$

Solution Steps

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}.

$$\frac{5\sqrt{3}-\sqrt{3}}{\sqrt{3}}-\sqrt{\frac{1}{5}}\sqrt{20}$$

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

$$\frac{4\sqrt{3}}{\sqrt{3}}-\sqrt{\frac{1}{5}}\sqrt{20}$$

Rationalize the denominator of \frac{4\sqrt{3}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{4\sqrt{3}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{1}{5}}\sqrt{20}$$

The square of \sqrt{3} is 3.

$$\frac{4\sqrt{3}\sqrt{3}}{3}-\sqrt{\frac{1}{5}}\sqrt{20}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{4\times 3}{3}-\sqrt{\frac{1}{5}}\sqrt{20}$$

Multiply 4 and 3 to get 12.

$$\frac{12}{3}-\sqrt{\frac{1}{5}}\sqrt{20}$$

Divide 12 by 3 to get 4.

$$4-\sqrt{\frac{1}{5}}\sqrt{20}$$

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

$$4-\frac{\sqrt{1}}{\sqrt{5}}\sqrt{20}$$

Calculate the square root of 1 and get 1.

$$4-\frac{1}{\sqrt{5}}\sqrt{20}$$

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

$$4-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\sqrt{20}$$

The square of \sqrt{5} is 5.

$$4-\frac{\sqrt{5}}{5}\sqrt{20}$$

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}.

$$4-\frac{\sqrt{5}}{5}\times 2\sqrt{5}$$

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

$$4-\frac{\sqrt{5}\times 2}{5}\sqrt{5}$$

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

$$4-\frac{\sqrt{5}\times 2\sqrt{5}}{5}$$

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

$$4-\frac{5\times 2}{5}$$

Cancel out 5 and 5.

$$4-2$$

Subtract 2 from 4 to get 2.

$$2$$

Factor

$$2$$