Answer
$$\frac{\sqrt{3}+\sqrt{4}}{\sqrt{13}-\sqrt{11}}=\frac{\sqrt{39}+\sqrt{33}+2\sqrt{13}+2\sqrt{11}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}+\sqrt{4}}{\sqrt{13}-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}+\sqrt{4}}{\sqrt{13}-\sqrt{11}}\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{39}+\sqrt{33}+2\sqrt{13}+2\sqrt{11}}{13+\sqrt{143}-\sqrt{143}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{39}+\sqrt{33}+2\sqrt{13}+2\sqrt{11}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{13} + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{3} + \sqrt{4}\right) } \cdot \left( \sqrt{13} + \sqrt{11}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{13}+\color{blue}{ \sqrt{3}} \cdot \sqrt{11}+\color{blue}{ \sqrt{4}} \cdot \sqrt{13}+\color{blue}{ \sqrt{4}} \cdot \sqrt{11} = \\ = \sqrt{39} + \sqrt{33} + 2 \sqrt{13} + 2 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{13}- \sqrt{11}\right) } \cdot \left( \sqrt{13} + \sqrt{11}\right) = \color{blue}{ \sqrt{13}} \cdot \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot \sqrt{13}\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 13 + \sqrt{143}- \sqrt{143}-11 $$ |
| ③ | Simplify numerator and denominator |
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