Answer
$$\frac{9}{\sqrt{9}+9\sqrt{7}}=\frac{-27+81\sqrt{7}}{558}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9}{\sqrt{9}+9\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9}{\sqrt{9}+9\sqrt{7}}\frac{\sqrt{9}-9\sqrt{7}}{\sqrt{9}-9\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{27-81\sqrt{7}}{9-27\sqrt{7}+27\sqrt{7}-567} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{27-81\sqrt{7}}{-558} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-27+81\sqrt{7}}{558}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{9}- 9 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 } \cdot \left( \sqrt{9}- 9 \sqrt{7}\right) = \color{blue}{9} \cdot \sqrt{9}+\color{blue}{9} \cdot- 9 \sqrt{7} = \\ = 27- 81 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{9} + 9 \sqrt{7}\right) } \cdot \left( \sqrt{9}- 9 \sqrt{7}\right) = \color{blue}{ \sqrt{9}} \cdot \sqrt{9}+\color{blue}{ \sqrt{9}} \cdot- 9 \sqrt{7}+\color{blue}{ 9 \sqrt{7}} \cdot \sqrt{9}+\color{blue}{ 9 \sqrt{7}} \cdot- 9 \sqrt{7} = \\ = 9- 27 \sqrt{7} + 27 \sqrt{7}-567 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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