Question
Answer
$$\dfrac{\sqrt{180}-2\sqrt{5}}{5\sqrt{5}+5}=\dfrac{5-\sqrt{5}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{180}-2\sqrt{5}}{5\sqrt{5}+5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{180}-2\sqrt{5}}{5\sqrt{5}+5}\frac{5\sqrt{5}-5}{5\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{150-30\sqrt{5}-50+10\sqrt{5}}{125-25\sqrt{5}+25\sqrt{5}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{100-20\sqrt{5}}{100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5-\sqrt{5}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{5}-5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{180}- 2 \sqrt{5}\right) } \cdot \left( 5 \sqrt{5}-5\right) = \color{blue}{ \sqrt{180}} \cdot 5 \sqrt{5}+\color{blue}{ \sqrt{180}} \cdot-5\color{blue}{- 2 \sqrt{5}} \cdot 5 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot-5 = \\ = 150- 30 \sqrt{5}-50 + 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{5} + 5\right) } \cdot \left( 5 \sqrt{5}-5\right) = \color{blue}{ 5 \sqrt{5}} \cdot 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot-5+\color{blue}{5} \cdot 5 \sqrt{5}+\color{blue}{5} \cdot-5 = \\ = 125- 25 \sqrt{5} + 25 \sqrt{5}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 20. |
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