Answer
$$\frac{\sqrt{7}-\sqrt{6}}{\sqrt{7}+\sqrt{6}}=13-2\sqrt{42}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{7}-\sqrt{6}}{\sqrt{7}+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}-\sqrt{6}}{\sqrt{7}+\sqrt{6}}\frac{\sqrt{7}-\sqrt{6}}{\sqrt{7}-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7-\sqrt{42}-\sqrt{42}+6}{7-\sqrt{42}+\sqrt{42}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{13-2\sqrt{42}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}13-2\sqrt{42}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7}- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{7}- \sqrt{6}\right) } \cdot \left( \sqrt{7}- \sqrt{6}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot- \sqrt{6}\color{blue}{- \sqrt{6}} \cdot \sqrt{7}\color{blue}{- \sqrt{6}} \cdot- \sqrt{6} = \\ = 7- \sqrt{42}- \sqrt{42} + 6 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7} + \sqrt{6}\right) } \cdot \left( \sqrt{7}- \sqrt{6}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot \sqrt{7}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 7- \sqrt{42} + \sqrt{42}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Remove 1 from denominator. |
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