Question
Answer
$$\dfrac{3}{6+9\sqrt{7}}=\dfrac{-2+3\sqrt{7}}{59}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{6+9\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{6+9\sqrt{7}}\frac{6-9\sqrt{7}}{6-9\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18-27\sqrt{7}}{36-54\sqrt{7}+54\sqrt{7}-567} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18-27\sqrt{7}}{-531} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2-3\sqrt{7}}{-59} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-2+3\sqrt{7}}{59}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 9 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 6- 9 \sqrt{7}\right) = \color{blue}{3} \cdot6+\color{blue}{3} \cdot- 9 \sqrt{7} = \\ = 18- 27 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 9 \sqrt{7}\right) } \cdot \left( 6- 9 \sqrt{7}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 9 \sqrt{7}+\color{blue}{ 9 \sqrt{7}} \cdot6+\color{blue}{ 9 \sqrt{7}} \cdot- 9 \sqrt{7} = \\ = 36- 54 \sqrt{7} + 54 \sqrt{7}-567 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Multiply both numerator and denominator by -1. |
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