Solve (2sqrt{6}+frac{sqrt{2}}{2})-(frac{sqrt{2}}{4}-sqrt{6})

$$( 2 \sqrt { 6 } + \frac { \sqrt { 2 } } { 2 } ) - ( \frac { \sqrt { 2 } } { 4 } - \sqrt { 6 } )$$

Evaluate

$$\frac{\sqrt{2}}{4}+3\sqrt{6}\approx 7.702022619$$

Short Solution Steps

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6} times \frac{4}{4}.

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

Since \frac{\sqrt{2}}{4} and \frac{4\sqrt{6}}{4} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

$$\frac{8\sqrt{6}+2\sqrt{2}-\sqrt{2}+4\sqrt{6}}{4}$$

Do the calculations in 8\sqrt{6}+2\sqrt{2}-\sqrt{2}+4\sqrt{6}.

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

Factor

$$\frac{\sqrt{2} + 12 \sqrt{6}}{4} = 7.702022618942808$$