Solve 1/sqrt{3+1/2}

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

Evaluate

$$\sqrt{3}-1\approx 0.732050808$$

Solution Steps

Divide 1 by \frac{\sqrt{3}+1}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{3}+1}{2}.

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

Rationalize the denominator of \frac{2}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

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

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

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

$$\sqrt{3}-1$$

Factor

$$\sqrt{3} - 1 = 0.732050808$$