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

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

Evaluate

$$1-2\sqrt{3}\approx -2.464101615$$

Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

To find the opposite of \sqrt{3}-1, find the opposite of each term.

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

The opposite of -1 is 1.

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

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

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