Question
Answer
$$\dfrac{14}{1+\sqrt{309}}=\dfrac{-1+\sqrt{309}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{14}{1+\sqrt{309}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{14}{1+\sqrt{309}}\frac{1-\sqrt{309}}{1-\sqrt{309}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{14-14\sqrt{309}}{1-\sqrt{309}+\sqrt{309}-309} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14-14\sqrt{309}}{-308} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1-\sqrt{309}}{-22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-1+\sqrt{309}}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1- \sqrt{309}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 14 } \cdot \left( 1- \sqrt{309}\right) = \color{blue}{14} \cdot1+\color{blue}{14} \cdot- \sqrt{309} = \\ = 14- 14 \sqrt{309} $$ Simplify denominator. $$ \color{blue}{ \left( 1 + \sqrt{309}\right) } \cdot \left( 1- \sqrt{309}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- \sqrt{309}+\color{blue}{ \sqrt{309}} \cdot1+\color{blue}{ \sqrt{309}} \cdot- \sqrt{309} = \\ = 1- \sqrt{309} + \sqrt{309}-309 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 14. |
| ⑤ | Multiply both numerator and denominator by -1. |
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