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

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

Evaluate

$$-4$$

Solution Steps

Anything divided by one gives itself.

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

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

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

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

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

Square \sqrt{3}. Square 2.

$$\sqrt{3}-2+\frac{\sqrt{3}+2}{3-4}$$

Subtract 4 from 3 to get -1.

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

Anything divided by -1 gives its opposite. To find the opposite of \sqrt{3}+2, find the opposite of each term.

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

Combine \sqrt{3} and -\sqrt{3} to get 0.

$$-2-2$$

Subtract 2 from -2 to get -4.

$$-4$$

Factor

$$-4$$