Solve frac{113-sqrt{11}}{sqrt{13}+sqrt{11}}+frac{sqrt{13}+sqrt{11}}{sqrt{13}-sqrt{11}}

$$\frac { 113 - \sqrt { 11 } } { \sqrt { 13 } + \sqrt { 11 } } + \frac { \sqrt { 13 } + \sqrt { 11 } } { \sqrt { 13 } - \sqrt { 11 } }$$

Evaluate

$$\frac{\sqrt{143}+113\sqrt{13}+35-113\sqrt{11}}{2}\approx 39.80347678$$

Short Solution Steps

Rationalize the denominator of \frac{113-\sqrt{11}}{\sqrt{13}+\sqrt{11}} by multiplying numerator and denominator by \sqrt{13}-\sqrt{11}.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

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}.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{\left(\sqrt{13}\right)^{2}-\left(\sqrt{11}\right)^{2}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{13-11}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

Subtract 11 from 13 to get 2.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{\left(\sqrt{13}-\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}$$

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}.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{\left(\sqrt{13}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{13-11}$$

Subtract 11 from 13 to get 2.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}+\sqrt{11}\right)\left(\sqrt{13}+\sqrt{11}\right)}{2}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}+\sqrt{11}\right)^{2}}{2}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{\left(\sqrt{13}\right)^{2}+2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}$$

The square of \sqrt{13} is 13.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{13+2\sqrt{13}\sqrt{11}+\left(\sqrt{11}\right)^{2}}{2}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{13+2\sqrt{143}+\left(\sqrt{11}\right)^{2}}{2}$$

The square of \sqrt{11} is 11.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{13+2\sqrt{143}+11}{2}$$

Add 13 and 11 to get 24.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{24+2\sqrt{143}}{2}$$

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

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+12+\sqrt{143}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12+\sqrt{143} times \frac{2}{2}.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2}+\frac{2\left(12+\sqrt{143}\right)}{2}$$

Since \frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)}{2} and \frac{2\left(12+\sqrt{143}\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{\left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)+2\left(12+\sqrt{143}\right)}{2}$$

Do the multiplications in \left(113-\sqrt{11}\right)\left(\sqrt{13}-\sqrt{11}\right)+2\left(12+\sqrt{143}\right).

$$\frac{113\sqrt{13}-113\sqrt{11}-\sqrt{143}+11+24+2\sqrt{143}}{2}$$

Do the calculations in 113\sqrt{13}-113\sqrt{11}-\sqrt{143}+11+24+2\sqrt{143}.

$$\frac{113\sqrt{13}-113\sqrt{11}+\sqrt{143}+35}{2}$$

Factor

$$\frac{\sqrt{143} + 113 \sqrt{13} + 35 - 113 \sqrt{11}}{2} = 39.803476780186$$