Solve 1/1+sqrt{3}

$$\frac { 1 } { 1 + \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{3}-1}{2}\approx 0.366025404$$

Solution Steps

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

$$\frac{1-\sqrt{3}}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}$$

Consider \left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1-\sqrt{3}}{1^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 1. Square \sqrt{3}.

$$\frac{1-\sqrt{3}}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{1-\sqrt{3}}{-2}$$

Multiply both numerator and denominator by -1.

$$\frac{-1+\sqrt{3}}{2}$$