Answer
$$\frac{8}{\sqrt{6}+5}=\frac{-8\sqrt{6}+40}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{\sqrt{6}+5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{\sqrt{6}+5}\frac{\sqrt{6}-5}{\sqrt{6}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8\sqrt{6}-40}{6-5\sqrt{6}+5\sqrt{6}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{6}-40}{-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-8\sqrt{6}+40}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{6}-5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( \sqrt{6}-5\right) = \color{blue}{8} \cdot \sqrt{6}+\color{blue}{8} \cdot-5 = \\ = 8 \sqrt{6}-40 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6} + 5\right) } \cdot \left( \sqrt{6}-5\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot-5+\color{blue}{5} \cdot \sqrt{6}+\color{blue}{5} \cdot-5 = \\ = 6- 5 \sqrt{6} + 5 \sqrt{6}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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