Question
Answer
$$\dfrac{8+\sqrt{14}}{6+\sqrt{14}}=\dfrac{17-\sqrt{14}}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8+\sqrt{14}}{6+\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8+\sqrt{14}}{6+\sqrt{14}}\frac{6-\sqrt{14}}{6-\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{48-8\sqrt{14}+6\sqrt{14}-14}{36-6\sqrt{14}+6\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{34-2\sqrt{14}}{22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{17-\sqrt{14}}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 8 + \sqrt{14}\right) } \cdot \left( 6- \sqrt{14}\right) = \color{blue}{8} \cdot6+\color{blue}{8} \cdot- \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot6+\color{blue}{ \sqrt{14}} \cdot- \sqrt{14} = \\ = 48- 8 \sqrt{14} + 6 \sqrt{14}-14 $$ Simplify denominator. $$ \color{blue}{ \left( 6 + \sqrt{14}\right) } \cdot \left( 6- \sqrt{14}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot6+\color{blue}{ \sqrt{14}} \cdot- \sqrt{14} = \\ = 36- 6 \sqrt{14} + 6 \sqrt{14}-14 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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