Question
Answer
$$\dfrac{7}{-9+\sqrt{5}}=\dfrac{-63-7\sqrt{5}}{76}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{-9+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{-9+\sqrt{5}}\frac{-9-\sqrt{5}}{-9-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-63-7\sqrt{5}}{81+9\sqrt{5}-9\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-63-7\sqrt{5}}{76}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -9- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( -9- \sqrt{5}\right) = \color{blue}{7} \cdot-9+\color{blue}{7} \cdot- \sqrt{5} = \\ = -63- 7 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( -9 + \sqrt{5}\right) } \cdot \left( -9- \sqrt{5}\right) = \color{blue}{-9} \cdot-9\color{blue}{-9} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-9+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 81 + 9 \sqrt{5}- 9 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
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