Question
Answer
$$\dfrac{\sqrt{180}+2\sqrt{5}}{5\sqrt{5}-5}=\dfrac{200+40\sqrt{5}}{100}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{180}+2\sqrt{5}}{5\sqrt{5}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{180}+2\sqrt{5}}{5\sqrt{5}-5}\frac{5\sqrt{5}+5}{5\sqrt{5}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{150+30\sqrt{5}+50+10\sqrt{5}}{125+25\sqrt{5}-25\sqrt{5}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{200+40\sqrt{5}}{100}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{5} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{180} + 2 \sqrt{5}\right) } \cdot \left( 5 \sqrt{5} + 5\right) = \color{blue}{ \sqrt{180}} \cdot 5 \sqrt{5}+\color{blue}{ \sqrt{180}} \cdot5+\color{blue}{ 2 \sqrt{5}} \cdot 5 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot5 = \\ = 150 + 30 \sqrt{5} + 50 + 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{5}-5\right) } \cdot \left( 5 \sqrt{5} + 5\right) = \color{blue}{ 5 \sqrt{5}} \cdot 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot5\color{blue}{-5} \cdot 5 \sqrt{5}\color{blue}{-5} \cdot5 = \\ = 125 + 25 \sqrt{5}- 25 \sqrt{5}-25 $$ |
| ③ | Simplify numerator and denominator |
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