Answer
$$\frac{10}{6+\sqrt{6}}=\frac{6-\sqrt{6}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{10}{6+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{10}{6+\sqrt{6}}\frac{6-\sqrt{6}}{6-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{60-10\sqrt{6}}{36-6\sqrt{6}+6\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60-10\sqrt{6}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6-\sqrt{6}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 10 } \cdot \left( 6- \sqrt{6}\right) = \color{blue}{10} \cdot6+\color{blue}{10} \cdot- \sqrt{6} = \\ = 60- 10 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 6 + \sqrt{6}\right) } \cdot \left( 6- \sqrt{6}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot6+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 36- 6 \sqrt{6} + 6 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 10. |
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