Answer
$$\frac{7-\sqrt{15}}{3+\sqrt{15}}=\frac{-18+5\sqrt{15}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7-\sqrt{15}}{3+\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7-\sqrt{15}}{3+\sqrt{15}}\frac{3-\sqrt{15}}{3-\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{21-7\sqrt{15}-3\sqrt{15}+15}{9-3\sqrt{15}+3\sqrt{15}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{36-10\sqrt{15}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{18-5\sqrt{15}}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-18+5\sqrt{15}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7- \sqrt{15}\right) } \cdot \left( 3- \sqrt{15}\right) = \color{blue}{7} \cdot3+\color{blue}{7} \cdot- \sqrt{15}\color{blue}{- \sqrt{15}} \cdot3\color{blue}{- \sqrt{15}} \cdot- \sqrt{15} = \\ = 21- 7 \sqrt{15}- 3 \sqrt{15} + 15 $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{15}\right) } \cdot \left( 3- \sqrt{15}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{15}+\color{blue}{ \sqrt{15}} \cdot3+\color{blue}{ \sqrt{15}} \cdot- \sqrt{15} = \\ = 9- 3 \sqrt{15} + 3 \sqrt{15}-15 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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