Solve (2sqrt{12}+1/5sqrt{75}-4sqrt{1/3})div1/2sqrt{3}=

$$( 2 \sqrt { 12 } + \frac { 1 } { 5 } \sqrt { 75 } - 4 \sqrt { \frac { 1 } { 3 } } ) \div \frac { 1 } { 2 } \sqrt { 3 } =$$

Evaluate

$$22$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{2\times 2\sqrt{3}+\frac{1}{5}\sqrt{75}-4\sqrt{\frac{1}{3}}}{\frac{1}{2}}\sqrt{3}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{3}+\frac{1}{5}\sqrt{75}-4\sqrt{\frac{1}{3}}}{\frac{1}{2}}\sqrt{3}$$

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}.

$$\frac{4\sqrt{3}+\frac{1}{5}\times 5\sqrt{3}-4\sqrt{\frac{1}{3}}}{\frac{1}{2}}\sqrt{3}$$

Cancel out 5 and 5.

$$\frac{4\sqrt{3}+\sqrt{3}-4\sqrt{\frac{1}{3}}}{\frac{1}{2}}\sqrt{3}$$

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

$$\frac{5\sqrt{3}-4\sqrt{\frac{1}{3}}}{\frac{1}{2}}\sqrt{3}$$

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

$$\frac{5\sqrt{3}-4\times \frac{\sqrt{1}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$\frac{5\sqrt{3}-4\times \frac{1}{\sqrt{3}}}{\frac{1}{2}}\sqrt{3}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{3}-4\times \frac{\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

Express -4\times \frac{\sqrt{3}}{3} as a single fraction.

$$\frac{5\sqrt{3}+\frac{-4\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

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

$$\frac{\frac{3\times 5\sqrt{3}}{3}+\frac{-4\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

Since \frac{3\times 5\sqrt{3}}{3} and \frac{-4\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3\times 5\sqrt{3}-4\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

Do the multiplications in 3\times 5\sqrt{3}-4\sqrt{3}.

$$\frac{\frac{15\sqrt{3}-4\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

Do the calculations in 15\sqrt{3}-4\sqrt{3}.

$$\frac{\frac{11\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{3}$$

Divide \frac{11\sqrt{3}}{3} by \frac{1}{2} by multiplying \frac{11\sqrt{3}}{3} by the reciprocal of \frac{1}{2}.

$$\frac{11\sqrt{3}\times 2}{3}\sqrt{3}$$

Multiply 11 and 2 to get 22.

$$\frac{22\sqrt{3}}{3}\sqrt{3}$$

Express \frac{22\sqrt{3}}{3}\sqrt{3} as a single fraction.

$$\frac{22\sqrt{3}\sqrt{3}}{3}$$

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

$$\frac{22\times 3}{3}$$

Cancel out 3 and 3.

$$22$$

Factor

$$2\times 11$$