Question
Answer
$$\dfrac{20}{5-\sqrt{15}}=\dfrac{100+20\sqrt{15}}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{20}{5-\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{20}{5-\sqrt{15}}\frac{5+\sqrt{15}}{5+\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{100+20\sqrt{15}}{25+5\sqrt{15}-5\sqrt{15}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{100+20\sqrt{15}}{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 20 } \cdot \left( 5 + \sqrt{15}\right) = \color{blue}{20} \cdot5+\color{blue}{20} \cdot \sqrt{15} = \\ = 100 + 20 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 5- \sqrt{15}\right) } \cdot \left( 5 + \sqrt{15}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{15}\color{blue}{- \sqrt{15}} \cdot5\color{blue}{- \sqrt{15}} \cdot \sqrt{15} = \\ = 25 + 5 \sqrt{15}- 5 \sqrt{15}-15 $$ |
| ③ | Simplify numerator and denominator |
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