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

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

Evaluate

$$3$$

Short Solution Steps

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

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

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

Do the multiplications in \sqrt{3}\left(\sqrt{3}+1\right)+\sqrt{3}\left(\sqrt{3}-1\right).

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

Do the calculations in 3+\sqrt{3}+3-\sqrt{3}.

$$\frac{6}{2}$$

Divide 6 by 2 to get 3.

$$3$$

Factor

$$3$$