Question
Answer
$$\dfrac{9}{12-3\sqrt{17}}=-\dfrac{108+27\sqrt{17}}{9}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9}{12-3\sqrt{17}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9}{12-3\sqrt{17}}\frac{12+3\sqrt{17}}{12+3\sqrt{17}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{108+27\sqrt{17}}{144+36\sqrt{17}-36\sqrt{17}-153} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{108+27\sqrt{17}}{-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{108+27\sqrt{17}}{9}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 12 + 3 \sqrt{17}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 } \cdot \left( 12 + 3 \sqrt{17}\right) = \color{blue}{9} \cdot12+\color{blue}{9} \cdot 3 \sqrt{17} = \\ = 108 + 27 \sqrt{17} $$ Simplify denominator. $$ \color{blue}{ \left( 12- 3 \sqrt{17}\right) } \cdot \left( 12 + 3 \sqrt{17}\right) = \color{blue}{12} \cdot12+\color{blue}{12} \cdot 3 \sqrt{17}\color{blue}{- 3 \sqrt{17}} \cdot12\color{blue}{- 3 \sqrt{17}} \cdot 3 \sqrt{17} = \\ = 144 + 36 \sqrt{17}- 36 \sqrt{17}-153 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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