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