Solve (frac{sqrt{13}-sqrt{11}}{sqrt{13}+sqrt{11}}+frac{sqrt{13}+sqrt{11}}{sqrt{13}-sqrt{11}})

$$( \frac { \sqrt { 13 } - \sqrt { 11 } } { \sqrt { 13 } + \sqrt { 11 } } + \frac { \sqrt { 13 } + \sqrt { 11 } } { \sqrt { 13 } - \sqrt { 11 } } )$$

Evaluate

$$24$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}} by multiplying numerator and denominator by \sqrt{13}-\sqrt{11}.

$$\frac{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Consider \left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{\left(\sqrt{13}\right)^{2}-\left(\sqrt{11}\right)^{2}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Square \sqrt{13}. Square \sqrt{11}.

$$\frac{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{13-11}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Subtract 11 from 13 to get 2.

$$\frac{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Multiply \sqrt{13}-\sqrt{11} and \sqrt{13}-\sqrt{11} to get \left(\sqrt{13}-\sqrt{11}\right)^{2}.

$$\frac{\left(\sqrt{13}-\sqrt{11}\right)^{2}}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{13}-\sqrt{11}\right)^{2}.

$$\frac{\left(\sqrt{13}\right)^{2}-2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

The square of \sqrt{13} is 13.

$$\frac{13-2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

To multiply \sqrt{13} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{13-2\sqrt{143}+\left(\sqrt{11}\right)^{2}}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

The square of \sqrt{11} is 11.

$$\frac{13-2\sqrt{143}+11}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Add 13 and 11 to get 24.

$$\frac{24-2\sqrt{143}}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Divide each term of 24-2\sqrt{143} by 2 to get 12-\sqrt{143}.

$$12-\sqrt{143}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Rationalize the denominator of \frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}} by multiplying numerator and denominator by \sqrt{13}+\sqrt{11}.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}$$

Consider \left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{\left(\sqrt{13}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$

Square \sqrt{13}. Square \sqrt{11}.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{13-11}$$

Subtract 11 from 13 to get 2.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{2}$$

Multiply \sqrt{13}+\sqrt{11} and \sqrt{13}+\sqrt{11} to get \left(\sqrt{13}+\sqrt{11}\right)^{2}.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}+\sqrt{11}\right)^{2}}{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{13}+\sqrt{11}\right)^{2}.

$$12-\sqrt{143}+\frac{\left(\sqrt{13}\right)^{2}+2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}$$

The square of \sqrt{13} is 13.

$$12-\sqrt{143}+\frac{13+2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}$$

To multiply \sqrt{13} and \sqrt{11}, multiply the numbers under the square root.

$$12-\sqrt{143}+\frac{13+2\sqrt{143}+\left(\sqrt{11}\right)^{2}}{2}$$

The square of \sqrt{11} is 11.

$$12-\sqrt{143}+\frac{13+2\sqrt{143}+11}{2}$$

Add 13 and 11 to get 24.

$$12-\sqrt{143}+\frac{24+2\sqrt{143}}{2}$$

Divide each term of 24+2\sqrt{143} by 2 to get 12+\sqrt{143}.

$$12-\sqrt{143}+12+\sqrt{143}$$

Add 12 and 12 to get 24.

$$24-\sqrt{143}+\sqrt{143}$$

Combine -\sqrt{143} and \sqrt{143} to get 0.

$$24$$

Factor

$$2^{3}\times 3$$