Question
Answer
$$\dfrac{-\sqrt{15}}{3+10\sqrt{3}}=\dfrac{\sqrt{15}-10\sqrt{5}}{97}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-\sqrt{15}}{3+10\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-\sqrt{15}}{3+10\sqrt{3}}\frac{3-10\sqrt{3}}{3-10\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-3\sqrt{15}+30\sqrt{5}}{9-30\sqrt{3}+30\sqrt{3}-300} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-3\sqrt{15}+30\sqrt{5}}{-291} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{15}+10\sqrt{5}}{-97} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{\sqrt{15}-10\sqrt{5}}{97}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- 10 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ - \sqrt{15} } \cdot \left( 3- 10 \sqrt{3}\right) = \color{blue}{- \sqrt{15}} \cdot3\color{blue}{- \sqrt{15}} \cdot- 10 \sqrt{3} = \\ = - 3 \sqrt{15} + 30 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + 10 \sqrt{3}\right) } \cdot \left( 3- 10 \sqrt{3}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 10 \sqrt{3}+\color{blue}{ 10 \sqrt{3}} \cdot3+\color{blue}{ 10 \sqrt{3}} \cdot- 10 \sqrt{3} = \\ = 9- 30 \sqrt{3} + 30 \sqrt{3}-300 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Multiply both numerator and denominator by -1. |
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