Solve frac{sqrt{3}-1}{2sqrt{2}}frac{7+24sqrt{3}}{50}-frac{24-7sqrt{3}}{50}frac{sqrt{3}+1}{2sqrt{2}}

$$\frac{ \sqrt{ 3 } -1 }{ 2 \sqrt{ 2 } } \frac{ 7+24 \sqrt{ 3 } }{ 50 } - \frac{ 24-7 \sqrt{ 3 } }{ 50 } \frac{ \sqrt{ 3 } +1 }{ 2 \sqrt{ 2 } }$$

Evaluate

$$\frac{31\sqrt{2}-17\sqrt{6}}{100}\approx 0.021992948$$

Short Solution Steps

Rationalize the denominator of \frac{\sqrt{3}-1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\times \frac{7+24\sqrt{3}}{50}-\frac{24-7\sqrt{3}}{50}\times \frac{\sqrt{3}+1}{2\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}}{2\times 2}\times \frac{7+24\sqrt{3}}{50}-\frac{24-7\sqrt{3}}{50}\times \frac{\sqrt{3}+1}{2\sqrt{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}}{4}\times \frac{7+24\sqrt{3}}{50}-\frac{24-7\sqrt{3}}{50}\times \frac{\sqrt{3}+1}{2\sqrt{2}}$$

Multiply \frac{\left(\sqrt{3}-1\right)\sqrt{2}}{4} times \frac{7+24\sqrt{3}}{50} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{24-7\sqrt{3}}{50}\times \frac{\sqrt{3}+1}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{3}+1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{24-7\sqrt{3}}{50}\times \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{24-7\sqrt{3}}{50}\times \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{24-7\sqrt{3}}{50}\times \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{4}$$

Multiply \frac{24-7\sqrt{3}}{50} times \frac{\left(\sqrt{3}+1\right)\sqrt{2}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}}{50\times 4}$$

Multiply 50 and 4 to get 200.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{4\times 50}-\frac{\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}}{200}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 4\times 50.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{200}-\frac{\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}}{200}$$

Since \frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)}{200} and \frac{\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}}{200} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)-\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}}{200}$$

Do the multiplications in \left(\sqrt{3}-1\right)\sqrt{2}\left(7+24\sqrt{3}\right)-\left(24-7\sqrt{3}\right)\left(\sqrt{3}+1\right)\sqrt{2}.

$$\frac{7\sqrt{6}+72\sqrt{2}-7\sqrt{2}-24\sqrt{6}-24\sqrt{6}-24\sqrt{2}+21\sqrt{2}+7\sqrt{6}}{200}$$

Do the calculations in 7\sqrt{6}+72\sqrt{2}-7\sqrt{2}-24\sqrt{6}-24\sqrt{6}-24\sqrt{2}+21\sqrt{2}+7\sqrt{6}.

$$\frac{-34\sqrt{6}+62\sqrt{2}}{200}$$

Factor

$$\frac{31 \sqrt{2} - 17 \sqrt{6}}{100} = 0.021992948062519276$$