Answer
$$\frac{5-\sqrt{6}}{5+\sqrt{6}}=\frac{31-10\sqrt{6}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5-\sqrt{6}}{5+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5-\sqrt{6}}{5+\sqrt{6}}\frac{5-\sqrt{6}}{5-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25-5\sqrt{6}-5\sqrt{6}+6}{25-5\sqrt{6}+5\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{31-10\sqrt{6}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5- \sqrt{6}\right) } \cdot \left( 5- \sqrt{6}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{6}\color{blue}{- \sqrt{6}} \cdot5\color{blue}{- \sqrt{6}} \cdot- \sqrt{6} = \\ = 25- 5 \sqrt{6}- 5 \sqrt{6} + 6 $$ Simplify denominator. $$ \color{blue}{ \left( 5 + \sqrt{6}\right) } \cdot \left( 5- \sqrt{6}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot5+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 25- 5 \sqrt{6} + 5 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
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