Question
Answer
$$\dfrac{1}{(2+5\sqrt{2})^2}=\dfrac{27-10\sqrt{2}}{1058}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{(2+5\sqrt{2})^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{4+10\sqrt{2}+10\sqrt{2}+50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{54+20\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{54+20\sqrt{2}}\frac{54-20\sqrt{2}}{54-20\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{54-20\sqrt{2}}{2916-1080\sqrt{2}+1080\sqrt{2}-800} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{54-20\sqrt{2}}{2116} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{27-10\sqrt{2}}{1058}\end{aligned} $$ | |
| ① | $$ (2+5\sqrt{2})^2 = \left( 2 + 5 \sqrt{2} \right) \cdot \left( 2 + 5 \sqrt{2} \right) = 4 + 10 \sqrt{2} + 10 \sqrt{2} + 50 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 54- 20 \sqrt{2}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 54- 20 \sqrt{2}\right) = \color{blue}{1} \cdot54+\color{blue}{1} \cdot- 20 \sqrt{2} = \\ = 54- 20 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 54 + 20 \sqrt{2}\right) } \cdot \left( 54- 20 \sqrt{2}\right) = \color{blue}{54} \cdot54+\color{blue}{54} \cdot- 20 \sqrt{2}+\color{blue}{ 20 \sqrt{2}} \cdot54+\color{blue}{ 20 \sqrt{2}} \cdot- 20 \sqrt{2} = \\ = 2916- 1080 \sqrt{2} + 1080 \sqrt{2}-800 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Divide both numerator and denominator by 2. |
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