Solve 12sqrt{1/6}-(sqrt{18}-4sqrt{3})div2sqrt{3}

$$12 \sqrt { \frac { 1 } { 6 } } - ( \sqrt { 18 } - 4 \sqrt { 3 } ) \div 2 \sqrt { 3 }$$

Evaluate

$$\frac{\sqrt{6}}{2}+6\approx 7.224744871$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$12\times \frac{\sqrt{1}}{\sqrt{6}}-\frac{\sqrt{18}-4\sqrt{3}}{2}\sqrt{3}$$

Calculate the square root of 1 and get 1.

$$12\times \frac{1}{\sqrt{6}}-\frac{\sqrt{18}-4\sqrt{3}}{2}\sqrt{3}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$12\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}-\frac{\sqrt{18}-4\sqrt{3}}{2}\sqrt{3}$$

The square of \sqrt{6} is 6.

$$12\times \frac{\sqrt{6}}{6}-\frac{\sqrt{18}-4\sqrt{3}}{2}\sqrt{3}$$

Cancel out 6, the greatest common factor in 12 and 6.

$$2\sqrt{6}-\frac{\sqrt{18}-4\sqrt{3}}{2}\sqrt{3}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$2\sqrt{6}-\frac{3\sqrt{2}-4\sqrt{3}}{2}\sqrt{3}$$

Express \frac{3\sqrt{2}-4\sqrt{3}}{2}\sqrt{3} as a single fraction.

$$2\sqrt{6}-\frac{\left(3\sqrt{2}-4\sqrt{3}\right)\sqrt{3}}{2}$$

Use the distributive property to multiply 3\sqrt{2}-4\sqrt{3} by \sqrt{3}.

$$2\sqrt{6}-\frac{3\sqrt{2}\sqrt{3}-4\left(\sqrt{3}\right)^{2}}{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2\sqrt{6}-\frac{3\sqrt{6}-4\left(\sqrt{3}\right)^{2}}{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}-\frac{3\sqrt{6}-4\times 3}{2}$$

Multiply -4 and 3 to get -12.

$$2\sqrt{6}-\frac{3\sqrt{6}-12}{2}$$

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

$$\frac{2\times 2\sqrt{6}}{2}-\frac{3\sqrt{6}-12}{2}$$

Since \frac{2\times 2\sqrt{6}}{2} and \frac{3\sqrt{6}-12}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 2\sqrt{6}-\left(3\sqrt{6}-12\right)}{2}$$

Do the multiplications in 2\times 2\sqrt{6}-\left(3\sqrt{6}-12\right).

$$\frac{4\sqrt{6}-3\sqrt{6}+12}{2}$$

Do the calculations in 4\sqrt{6}-3\sqrt{6}+12.

$$\frac{\sqrt{6}+12}{2}$$

Factor

$$\frac{\sqrt{6} + 12}{2} = 7.224744871391589$$