Answer
$$\frac{1}{8+5\sqrt{2}}=\frac{8-5\sqrt{2}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{8+5\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{8+5\sqrt{2}}\frac{8-5\sqrt{2}}{8-5\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8-5\sqrt{2}}{64-40\sqrt{2}+40\sqrt{2}-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8-5\sqrt{2}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8- 5 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 8- 5 \sqrt{2}\right) = \color{blue}{1} \cdot8+\color{blue}{1} \cdot- 5 \sqrt{2} = \\ = 8- 5 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 8 + 5 \sqrt{2}\right) } \cdot \left( 8- 5 \sqrt{2}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot- 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot8+\color{blue}{ 5 \sqrt{2}} \cdot- 5 \sqrt{2} = \\ = 64- 40 \sqrt{2} + 40 \sqrt{2}-50 $$ |
| ③ | Simplify numerator and denominator |
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