Solve (2)sqrt{3}(1-sqrt{15})-3sqrt{1/5}

$$( 2 ) \sqrt { 3 } ( 1 - \sqrt { 15 } ) - 3 \sqrt { \frac { 1 } { 5 } }$$

Evaluate

$$-\frac{33\sqrt{5}}{5}+2\sqrt{3}\approx -11.293947036$$

Short Solution Steps

Use the distributive property to multiply 2\sqrt{3} by 1-\sqrt{15}.

$$2\sqrt{3}-2\sqrt{3}\sqrt{15}-3\sqrt{\frac{1}{5}}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

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

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

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

Multiply -2 and 3 to get -6.

$$2\sqrt{3}-6\sqrt{5}-3\sqrt{\frac{1}{5}}$$

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

$$2\sqrt{3}-6\sqrt{5}-3\times \frac{\sqrt{1}}{\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}-6\sqrt{5}-3\times \frac{1}{\sqrt{5}}$$

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

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

The square of \sqrt{5} is 5.

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

Express -3\times \frac{\sqrt{5}}{5} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3}-6\sqrt{5} times \frac{5}{5}.

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

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

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

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

$$\frac{10\sqrt{3}-30\sqrt{5}-3\sqrt{5}}{5}$$

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

$$\frac{10\sqrt{3}-33\sqrt{5}}{5}$$