Answer
$$\frac{1}{2\sqrt{2}+6\sqrt{6}}=\frac{-\sqrt{2}+3\sqrt{6}}{104}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2\sqrt{2}+6\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2\sqrt{2}+6\sqrt{6}}\frac{2\sqrt{2}-6\sqrt{6}}{2\sqrt{2}-6\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{2}-6\sqrt{6}}{8-24\sqrt{3}+24\sqrt{3}-216} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{2}-6\sqrt{6}}{-208} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}-3\sqrt{6}}{-104} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-\sqrt{2}+3\sqrt{6}}{104}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{2}- 6 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 2 \sqrt{2}- 6 \sqrt{6}\right) = \color{blue}{1} \cdot 2 \sqrt{2}+\color{blue}{1} \cdot- 6 \sqrt{6} = \\ = 2 \sqrt{2}- 6 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{2} + 6 \sqrt{6}\right) } \cdot \left( 2 \sqrt{2}- 6 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{2}} \cdot 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot- 6 \sqrt{6}+\color{blue}{ 6 \sqrt{6}} \cdot 2 \sqrt{2}+\color{blue}{ 6 \sqrt{6}} \cdot- 6 \sqrt{6} = \\ = 8- 24 \sqrt{3} + 24 \sqrt{3}-216 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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