Solve 3/1+sqrt{3}

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

Evaluate

$$\frac{3\left(\sqrt{3}-1\right)}{2}\approx 1.098076211$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Use the distributive property to multiply 3 by 1-\sqrt{3}.

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