Solve 2/sqrt{3-1}+sqrt{3}

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

Evaluate

$$2\sqrt{3}+1\approx 4.464101615$$

Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

Combine \sqrt{3} and \sqrt{3} to get 2\sqrt{3}.

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