Solve 2sqrt{6}+frac{3-sqrt{5}}{2}-4/sqrt{3}

$$2 \sqrt { 6 } + \frac { 3 - \sqrt { 5 } } { 2 } - \frac { 4 } { \sqrt { 3 } }$$

Evaluate

$$-\frac{\sqrt{5}}{2}-\frac{4\sqrt{3}}{3}+2\sqrt{6}+\frac{3}{2}\approx 2.97154442$$

Short Solution Steps

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

$$2\sqrt{6}+\frac{3-\sqrt{5}}{2}-\frac{4\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}+\frac{3-\sqrt{5}}{2}-\frac{4\sqrt{3}}{3}$$

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

$$\frac{2\times 2\sqrt{6}}{2}+\frac{3-\sqrt{5}}{2}-\frac{4\sqrt{3}}{3}$$

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

$$\frac{2\times 2\sqrt{6}+3-\sqrt{5}}{2}-\frac{4\sqrt{3}}{3}$$

Do the multiplications in 2\times 2\sqrt{6}+3-\sqrt{5}.

$$\frac{4\sqrt{6}+3-\sqrt{5}}{2}-\frac{4\sqrt{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{4\sqrt{6}+3-\sqrt{5}}{2} times \frac{3}{3}. Multiply \frac{4\sqrt{3}}{3} times \frac{2}{2}.

$$\frac{3\left(4\sqrt{6}+3-\sqrt{5}\right)}{6}-\frac{2\times 4\sqrt{3}}{6}$$

Since \frac{3\left(4\sqrt{6}+3-\sqrt{5}\right)}{6} and \frac{2\times 4\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(4\sqrt{6}+3-\sqrt{5}\right)-2\times 4\sqrt{3}}{6}$$

Do the multiplications in 3\left(4\sqrt{6}+3-\sqrt{5}\right)-2\times 4\sqrt{3}.

$$\frac{12\sqrt{6}+9-3\sqrt{5}-8\sqrt{3}}{6}$$

Factor

$$\frac{12 \sqrt{6} + 9 - 3 \sqrt{5} - 8 \sqrt{3}}{6} = 2.9715444200579575$$