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

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

Evaluate

$$2$$

Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

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

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

The opposite of -1 is 1.

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

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

$$1+1$$

Add 1 and 1 to get 2.

$$2$$

Factor

$$2$$