Solve frac{2-sqrt{3}}{sqrt{3}-1}

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

Evaluate

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

Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Apply the distributive property by multiplying each term of 2-\sqrt{3} by each term of \sqrt{3}+1.

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 2 to get -1.

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

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

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