Question
Answer
$$\dfrac{4}{-3+\sqrt{11}}=\dfrac{12+4\sqrt{11}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{-3+\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{-3+\sqrt{11}}\frac{-3-\sqrt{11}}{-3-\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-12-4\sqrt{11}}{9+3\sqrt{11}-3\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-12-4\sqrt{11}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{12+4\sqrt{11}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -3- \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( -3- \sqrt{11}\right) = \color{blue}{4} \cdot-3+\color{blue}{4} \cdot- \sqrt{11} = \\ = -12- 4 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( -3 + \sqrt{11}\right) } \cdot \left( -3- \sqrt{11}\right) = \color{blue}{-3} \cdot-3\color{blue}{-3} \cdot- \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot-3+\color{blue}{ \sqrt{11}} \cdot- \sqrt{11} = \\ = 9 + 3 \sqrt{11}- 3 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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