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

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

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

Solution Steps

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

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

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

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

Square 2. Square \sqrt{3}.

$$\frac{2+\sqrt{3}}{4-3}-\frac{1}{2+\sqrt{3}}$$

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

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

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

To find the opposite of 2-\sqrt{3}, find the opposite of each term.

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

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

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

Subtract 2 from 2 to get 0.

$$\sqrt{3}+\sqrt{3}$$

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

$$2\sqrt{3}$$