Question
Answer
$$\dfrac{-3+5\sqrt{5}}{2+2\sqrt{5}}=\dfrac{7-2\sqrt{5}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-3+5\sqrt{5}}{2+2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-3+5\sqrt{5}}{2+2\sqrt{5}}\frac{2-2\sqrt{5}}{2-2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-6+6\sqrt{5}+10\sqrt{5}-50}{4-4\sqrt{5}+4\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-56+16\sqrt{5}}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-7+2\sqrt{5}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{7-2\sqrt{5}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( -3 + 5 \sqrt{5}\right) } \cdot \left( 2- 2 \sqrt{5}\right) = \color{blue}{-3} \cdot2\color{blue}{-3} \cdot- 2 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot2+\color{blue}{ 5 \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = -6 + 6 \sqrt{5} + 10 \sqrt{5}-50 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + 2 \sqrt{5}\right) } \cdot \left( 2- 2 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot2+\color{blue}{ 2 \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = 4- 4 \sqrt{5} + 4 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 8. |
| ⑤ | Multiply both numerator and denominator by -1. |
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