Question
Answer
$$\dfrac{\sqrt{5}}{5\sqrt{3}-\sqrt{5}}=\dfrac{\sqrt{15}+1}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{5\sqrt{3}-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{5\sqrt{3}-\sqrt{5}}\frac{5\sqrt{3}+\sqrt{5}}{5\sqrt{3}+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{15}+5}{75+5\sqrt{15}-5\sqrt{15}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{15}+5}{70} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{15}+1}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3} + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( 5 \sqrt{3} + \sqrt{5}\right) = \color{blue}{ \sqrt{5}} \cdot 5 \sqrt{3}+\color{blue}{ \sqrt{5}} \cdot \sqrt{5} = \\ = 5 \sqrt{15} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3}- \sqrt{5}\right) } \cdot \left( 5 \sqrt{3} + \sqrt{5}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot 5 \sqrt{3}\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 75 + 5 \sqrt{15}- 5 \sqrt{15}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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