Question
Answer
$$\dfrac{140}{-17+15\sqrt{23}}=\dfrac{2380+2100\sqrt{23}}{4886}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{140}{-17+15\sqrt{23}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{140}{-17+15\sqrt{23}}\frac{-17-15\sqrt{23}}{-17-15\sqrt{23}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-2380-2100\sqrt{23}}{289+255\sqrt{23}-255\sqrt{23}-5175} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2380-2100\sqrt{23}}{-4886} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2380+2100\sqrt{23}}{4886}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -17- 15 \sqrt{23}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 140 } \cdot \left( -17- 15 \sqrt{23}\right) = \color{blue}{140} \cdot-17+\color{blue}{140} \cdot- 15 \sqrt{23} = \\ = -2380- 2100 \sqrt{23} $$ Simplify denominator. $$ \color{blue}{ \left( -17 + 15 \sqrt{23}\right) } \cdot \left( -17- 15 \sqrt{23}\right) = \color{blue}{-17} \cdot-17\color{blue}{-17} \cdot- 15 \sqrt{23}+\color{blue}{ 15 \sqrt{23}} \cdot-17+\color{blue}{ 15 \sqrt{23}} \cdot- 15 \sqrt{23} = \\ = 289 + 255 \sqrt{23}- 255 \sqrt{23}-5175 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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