Question
Answer
$$\dfrac{4+\sqrt{6}}{5-\sqrt{122}}=-\dfrac{20+4\sqrt{122}+5\sqrt{6}+2\sqrt{183}}{97}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+\sqrt{6}}{5-\sqrt{122}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+\sqrt{6}}{5-\sqrt{122}}\frac{5+\sqrt{122}}{5+\sqrt{122}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20+4\sqrt{122}+5\sqrt{6}+2\sqrt{183}}{25+5\sqrt{122}-5\sqrt{122}-122} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20+4\sqrt{122}+5\sqrt{6}+2\sqrt{183}}{-97} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{20+4\sqrt{122}+5\sqrt{6}+2\sqrt{183}}{97}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + \sqrt{122}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + \sqrt{6}\right) } \cdot \left( 5 + \sqrt{122}\right) = \color{blue}{4} \cdot5+\color{blue}{4} \cdot \sqrt{122}+\color{blue}{ \sqrt{6}} \cdot5+\color{blue}{ \sqrt{6}} \cdot \sqrt{122} = \\ = 20 + 4 \sqrt{122} + 5 \sqrt{6} + 2 \sqrt{183} $$ Simplify denominator. $$ \color{blue}{ \left( 5- \sqrt{122}\right) } \cdot \left( 5 + \sqrt{122}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{122}\color{blue}{- \sqrt{122}} \cdot5\color{blue}{- \sqrt{122}} \cdot \sqrt{122} = \\ = 25 + 5 \sqrt{122}- 5 \sqrt{122}-122 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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