Answer
$$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=4+\sqrt{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+\sqrt{15}+\sqrt{15}+3}{5+\sqrt{15}-\sqrt{15}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8+2\sqrt{15}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4+\sqrt{15}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4+\sqrt{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{3}\right) } \cdot \left( \sqrt{5} + \sqrt{3}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot \sqrt{5}+\color{blue}{ \sqrt{3}} \cdot \sqrt{3} = \\ = 5 + \sqrt{15} + \sqrt{15} + 3 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}- \sqrt{3}\right) } \cdot \left( \sqrt{5} + \sqrt{3}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot \sqrt{5}\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 5 + \sqrt{15}- \sqrt{15}-3 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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