Question
Answer
$$\dfrac{\sqrt{2}+5}{\sqrt{2}-5}=-\dfrac{27+10\sqrt{2}}{23}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}+5}{\sqrt{2}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}+5}{\sqrt{2}-5}\frac{\sqrt{2}+5}{\sqrt{2}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2+5\sqrt{2}+5\sqrt{2}+25}{2+5\sqrt{2}-5\sqrt{2}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{27+10\sqrt{2}}{-23} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{27+10\sqrt{2}}{23}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2} + 5\right) } \cdot \left( \sqrt{2} + 5\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot5+\color{blue}{5} \cdot \sqrt{2}+\color{blue}{5} \cdot5 = \\ = 2 + 5 \sqrt{2} + 5 \sqrt{2} + 25 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2}-5\right) } \cdot \left( \sqrt{2} + 5\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot5\color{blue}{-5} \cdot \sqrt{2}\color{blue}{-5} \cdot5 = \\ = 2 + 5 \sqrt{2}- 5 \sqrt{2}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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