Question
Answer
$$\dfrac{\sqrt{2}}{\sqrt{5}-8}=-\dfrac{\sqrt{10}+8\sqrt{2}}{59}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{\sqrt{5}-8}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{\sqrt{5}-8}\frac{\sqrt{5}+8}{\sqrt{5}+8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{10}+8\sqrt{2}}{5+8\sqrt{5}-8\sqrt{5}-64} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{10}+8\sqrt{2}}{-59} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{\sqrt{10}+8\sqrt{2}}{59}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 8} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( \sqrt{5} + 8\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot8 = \\ = \sqrt{10} + 8 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-8\right) } \cdot \left( \sqrt{5} + 8\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot8\color{blue}{-8} \cdot \sqrt{5}\color{blue}{-8} \cdot8 = \\ = 5 + 8 \sqrt{5}- 8 \sqrt{5}-64 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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