Solve 1/1+sqrt{11}

$$\frac { 1 } { 1 + \sqrt { 11 } }$$

Evaluate

$$\frac{\sqrt{11}-1}{10}\approx 0.231662479$$

Solution Steps

Rationalize the denominator of \frac{1}{1+\sqrt{11}} by multiplying numerator and denominator by 1-\sqrt{11}.

$$\frac{1-\sqrt{11}}{\left(1+\sqrt{11}\right)\left(1-\sqrt{11}\right)}$$

Consider \left(1+\sqrt{11}\right)\left(1-\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1-\sqrt{11}}{1^{2}-\left(\sqrt{11}\right)^{2}}$$

Square 1. Square \sqrt{11}.

$$\frac{1-\sqrt{11}}{1-11}$$

Subtract 11 from 1 to get -10.

$$\frac{1-\sqrt{11}}{-10}$$

Multiply both numerator and denominator by -1.

$$\frac{-1+\sqrt{11}}{10}$$