Solve frac{left(frac{7+sqrt{21}}{2}+frac{7-sqrt{21}}{2}right)left(frac{7+sqrt{21}}{2}-frac{7-sqrt{21}}{2}right)}{2}

$$\frac{ \left( \frac{ 7+ \sqrt{ 21 } }{ 2 } + \frac{ 7- \sqrt{ 21 } }{ 2 } \right) \left( \frac{ 7+ \sqrt{ 21 } }{ 2 } - \frac{ 7- \sqrt{ 21 } }{ 2 } \right) }{ 2 }$$

Evaluate

$$\frac{7\sqrt{21}}{2}\approx 16.039014932$$

Short Solution Steps

Since \frac{7+\sqrt{21}}{2} and \frac{7-\sqrt{21}}{2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{7+\sqrt{21}+7-\sqrt{21}}{2}\left(\frac{7+\sqrt{21}}{2}-\frac{7-\sqrt{21}}{2}\right)}{2}$$

Do the calculations in 7+\sqrt{21}+7-\sqrt{21}.

$$\frac{\frac{14}{2}\left(\frac{7+\sqrt{21}}{2}-\frac{7-\sqrt{21}}{2}\right)}{2}$$

Since \frac{7+\sqrt{21}}{2} and \frac{7-\sqrt{21}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{14}{2}\times \frac{7+\sqrt{21}-\left(7-\sqrt{21}\right)}{2}}{2}$$

Do the multiplications in 7+\sqrt{21}-\left(7-\sqrt{21}\right).

$$\frac{\frac{14}{2}\times \frac{7+\sqrt{21}-7+\sqrt{21}}{2}}{2}$$

Do the calculations in 7+\sqrt{21}-7+\sqrt{21}.

$$\frac{\frac{14}{2}\times \frac{2\sqrt{21}}{2}}{2}$$

Cancel out 2 and 2.

$$\frac{\frac{14}{2}\sqrt{21}}{2}$$

Express \frac{14}{2}\sqrt{21} as a single fraction.

$$\frac{\frac{14\sqrt{21}}{2}}{2}$$

Express \frac{\frac{14\sqrt{21}}{2}}{2} as a single fraction.

$$\frac{14\sqrt{21}}{2\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{7\sqrt{21}}{2}$$