Answer
$$\frac{9}{9-9\sqrt{7}}=-\frac{1+\sqrt{7}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9}{9-9\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9}{9-9\sqrt{7}}\frac{9+9\sqrt{7}}{9+9\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{81+81\sqrt{7}}{81+81\sqrt{7}-81\sqrt{7}-567} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{81+81\sqrt{7}}{-486} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\sqrt{7}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{1+\sqrt{7}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9 + 9 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 } \cdot \left( 9 + 9 \sqrt{7}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot 9 \sqrt{7} = \\ = 81 + 81 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 9- 9 \sqrt{7}\right) } \cdot \left( 9 + 9 \sqrt{7}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot 9 \sqrt{7}\color{blue}{- 9 \sqrt{7}} \cdot9\color{blue}{- 9 \sqrt{7}} \cdot 9 \sqrt{7} = \\ = 81 + 81 \sqrt{7}- 81 \sqrt{7}-567 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 81. |
| ⑤ | Place a negative sign in front of a fraction. |
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