Question
Answer
$$\dfrac{-5}{5+\sqrt{20}}=-5+2\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-5}{5+\sqrt{20}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-5}{5+\sqrt{20}}\frac{5-\sqrt{20}}{5-\sqrt{20}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-25+10\sqrt{5}}{25-10\sqrt{5}+10\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-25+10\sqrt{5}}{5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-5+2\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-5+2\sqrt{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- \sqrt{20}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ -5 } \cdot \left( 5- \sqrt{20}\right) = \color{blue}{-5} \cdot5\color{blue}{-5} \cdot- \sqrt{20} = \\ = -25 + 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + \sqrt{20}\right) } \cdot \left( 5- \sqrt{20}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{20}+\color{blue}{ \sqrt{20}} \cdot5+\color{blue}{ \sqrt{20}} \cdot- \sqrt{20} = \\ = 25- 10 \sqrt{5} + 10 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Remove 1 from denominator. |
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