Question
Answer
$$\dfrac{\sqrt{2}}{8+8\sqrt{7}}=\dfrac{-\sqrt{2}+\sqrt{14}}{48}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{8+8\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{8+8\sqrt{7}}\frac{8-8\sqrt{7}}{8-8\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8\sqrt{2}-8\sqrt{14}}{64-64\sqrt{7}+64\sqrt{7}-448} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{2}-8\sqrt{14}}{-384} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}-\sqrt{14}}{-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-\sqrt{2}+\sqrt{14}}{48}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8- 8 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( 8- 8 \sqrt{7}\right) = \color{blue}{ \sqrt{2}} \cdot8+\color{blue}{ \sqrt{2}} \cdot- 8 \sqrt{7} = \\ = 8 \sqrt{2}- 8 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 8 + 8 \sqrt{7}\right) } \cdot \left( 8- 8 \sqrt{7}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot- 8 \sqrt{7}+\color{blue}{ 8 \sqrt{7}} \cdot8+\color{blue}{ 8 \sqrt{7}} \cdot- 8 \sqrt{7} = \\ = 64- 64 \sqrt{7} + 64 \sqrt{7}-448 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 8. |
| ⑤ | Multiply both numerator and denominator by -1. |
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