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

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

Evaluate

$$2$$

Short Solution Steps

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

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

Since -\frac{1}{2} and \frac{\sqrt{3}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Divide 1 by \frac{-1-\sqrt{3}}{2} by multiplying 1 by the reciprocal of \frac{-1-\sqrt{3}}{2}.

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

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

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

Consider \left(-1-\sqrt{3}\right)\left(-1+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square -1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Cancel out -2 and -2.

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

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

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

Since -\frac{1}{2} and \frac{\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

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

Divide 1 by \frac{-1+\sqrt{3}}{2} by multiplying 1 by the reciprocal of \frac{-1+\sqrt{3}}{2}.

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

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

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

Consider \left(-1+\sqrt{3}\right)\left(-1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square -1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Cancel out -2 and -2.

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

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

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

The opposite of -1 is 1.

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

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

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

The opposite of -1 is 1.

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

The opposite of -\sqrt{3} is \sqrt{3}.

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

Add 1 and 1 to get 2.

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

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

$$2$$

Factor

$$2$$