Solve frac{(4-sqrt{5})(4+sqrt{5})}{2sqrt{11}}

$$\frac { ( 4 - \sqrt { 5 } ) ( 4 + \sqrt { 5 } ) } { 2 \sqrt { 11 } }$$

Evaluate

$$\frac{\sqrt{11}}{2}\approx 1.658312395$$

Solution Steps

Consider \left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4^{2}-\left(\sqrt{5}\right)^{2}}{2\sqrt{11}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16-\left(\sqrt{5}\right)^{2}}{2\sqrt{11}}$$

The square of \sqrt{5} is 5.

$$\frac{16-5}{2\sqrt{11}}$$

Subtract 5 from 16 to get 11.

$$\frac{11}{2\sqrt{11}}$$

Rationalize the denominator of \frac{11}{2\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{11\sqrt{11}}{2\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{11\sqrt{11}}{2\times 11}$$

Cancel out 11 in both numerator and denominator.

$$\frac{\sqrt{11}}{2}$$