Solve (sqrt{24}-sqrt{1/2})-(1/8+sqrt{6})

$$( \sqrt { 24 } - \sqrt { \frac { 1 } { 2 } } ) - ( \frac { 1 } { 8 } + \sqrt { 6 } )$$

Evaluate

$$-\frac{\sqrt{2}}{2}+\sqrt{6}-\frac{1}{8}\approx 1.617382962$$

Short Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$2\sqrt{6}-\sqrt{\frac{1}{2}}-\left(\frac{1}{8}+\sqrt{6}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$2\sqrt{6}-\frac{\sqrt{1}}{\sqrt{2}}-\left(\frac{1}{8}+\sqrt{6}\right)$$

Calculate the square root of 1 and get 1.

$$2\sqrt{6}-\frac{1}{\sqrt{2}}-\left(\frac{1}{8}+\sqrt{6}\right)$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\sqrt{6}-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\left(\frac{1}{8}+\sqrt{6}\right)$$

The square of \sqrt{2} is 2.

$$2\sqrt{6}-\frac{\sqrt{2}}{2}-\left(\frac{1}{8}+\sqrt{6}\right)$$

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

$$\frac{2\times 2\sqrt{6}}{2}-\frac{\sqrt{2}}{2}-\left(\frac{1}{8}+\sqrt{6}\right)$$

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

$$\frac{2\times 2\sqrt{6}-\sqrt{2}}{2}-\left(\frac{1}{8}+\sqrt{6}\right)$$

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

$$\frac{4\sqrt{6}-\sqrt{2}}{2}-\left(\frac{1}{8}+\sqrt{6}\right)$$

To find the opposite of \frac{1}{8}+\sqrt{6}, find the opposite of each term.

$$\frac{4\sqrt{6}-\sqrt{2}}{2}-\frac{1}{8}-\sqrt{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 8 is 8. Multiply \frac{4\sqrt{6}-\sqrt{2}}{2} times \frac{4}{4}.

$$\frac{4\left(4\sqrt{6}-\sqrt{2}\right)}{8}-\frac{1}{8}-\sqrt{6}$$

Since \frac{4\left(4\sqrt{6}-\sqrt{2}\right)}{8} and \frac{1}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(4\sqrt{6}-\sqrt{2}\right)-1}{8}-\sqrt{6}$$

Do the multiplications in 4\left(4\sqrt{6}-\sqrt{2}\right)-1.

$$\frac{16\sqrt{6}-4\sqrt{2}-1}{8}-\sqrt{6}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6} times \frac{8}{8}.

$$\frac{16\sqrt{6}-4\sqrt{2}-1}{8}-\frac{8\sqrt{6}}{8}$$

Since \frac{16\sqrt{6}-4\sqrt{2}-1}{8} and \frac{8\sqrt{6}}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{16\sqrt{6}-4\sqrt{2}-1-8\sqrt{6}}{8}$$

Do the calculations in 16\sqrt{6}-4\sqrt{2}-1-8\sqrt{6}.

$$\frac{8\sqrt{6}-4\sqrt{2}-1}{8}$$

Factor

$$\frac{8 \sqrt{6} - 4 \sqrt{2} - 1}{8} = 1.6173829615966304$$