Answer
$$\frac{-3\sqrt{15}-5-16\sqrt{3}+4\sqrt{5}}{\sqrt{3}+\sqrt{5}}=2\sqrt{5}-5\sqrt{3}+34-10\sqrt{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-3\sqrt{15}-5-16\sqrt{3}+4\sqrt{5}}{\sqrt{3}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-3\sqrt{15}-5-16\sqrt{3}+4\sqrt{5}}{\sqrt{3}+\sqrt{5}}\frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-9\sqrt{5}+15\sqrt{3}-5\sqrt{3}+5\sqrt{5}-48+16\sqrt{15}+4\sqrt{15}-20}{3-\sqrt{15}+\sqrt{15}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-4\sqrt{5}+10\sqrt{3}-68+20\sqrt{15}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-2\sqrt{5}+5\sqrt{3}-34+10\sqrt{15}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{2\sqrt{5}-5\sqrt{3}+34-10\sqrt{15}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}2\sqrt{5}-5\sqrt{3}+34-10\sqrt{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( - 3 \sqrt{15}-5- 16 \sqrt{3} + 4 \sqrt{5}\right) } \cdot \left( \sqrt{3}- \sqrt{5}\right) = \color{blue}{- 3 \sqrt{15}} \cdot \sqrt{3}\color{blue}{- 3 \sqrt{15}} \cdot- \sqrt{5}\color{blue}{-5} \cdot \sqrt{3}\color{blue}{-5} \cdot- \sqrt{5}\color{blue}{- 16 \sqrt{3}} \cdot \sqrt{3}\color{blue}{- 16 \sqrt{3}} \cdot- \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ 4 \sqrt{5}} \cdot- \sqrt{5} = \\ = - 9 \sqrt{5} + 15 \sqrt{3}- 5 \sqrt{3} + 5 \sqrt{5}-48 + 16 \sqrt{15} + 4 \sqrt{15}-20 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + \sqrt{5}\right) } \cdot \left( \sqrt{3}- \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 3- \sqrt{15} + \sqrt{15}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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