Solve frac{(sqrt{3}+1)}{(sqrt{3}-2)}

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

Evaluate

$$-3\sqrt{3}-5\approx -10.196152423$$

Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 2.

$$\frac{\left(\sqrt{3}+1\right)\left(\sqrt{3}+2\right)}{3-4}$$

Subtract 4 from 3 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Add 3 and 2 to get 5.

$$-\left(5+3\sqrt{3}\right)$$

To find the opposite of 5+3\sqrt{3}, find the opposite of each term.

$$-5-3\sqrt{3}$$