Question
Answer
$$\dfrac{45+22\sqrt{6}}{4+2\sqrt{6}}=\dfrac{42+\sqrt{6}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{45+22\sqrt{6}}{4+2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{45+22\sqrt{6}}{4+2\sqrt{6}}\frac{4-2\sqrt{6}}{4-2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{180-90\sqrt{6}+88\sqrt{6}-264}{16-8\sqrt{6}+8\sqrt{6}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-84-2\sqrt{6}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-42-\sqrt{6}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{42+\sqrt{6}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 45 + 22 \sqrt{6}\right) } \cdot \left( 4- 2 \sqrt{6}\right) = \color{blue}{45} \cdot4+\color{blue}{45} \cdot- 2 \sqrt{6}+\color{blue}{ 22 \sqrt{6}} \cdot4+\color{blue}{ 22 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 180- 90 \sqrt{6} + 88 \sqrt{6}-264 $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 2 \sqrt{6}\right) } \cdot \left( 4- 2 \sqrt{6}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 2 \sqrt{6}+\color{blue}{ 2 \sqrt{6}} \cdot4+\color{blue}{ 2 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 16- 8 \sqrt{6} + 8 \sqrt{6}-24 $$ |
| 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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