Question
Answer
$$\dfrac{3-7\sqrt{2}}{3+7\sqrt{2}}=\dfrac{-107+42\sqrt{2}}{89}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-7\sqrt{2}}{3+7\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3-7\sqrt{2}}{3+7\sqrt{2}}\frac{3-7\sqrt{2}}{3-7\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9-21\sqrt{2}-21\sqrt{2}+98}{9-21\sqrt{2}+21\sqrt{2}-98} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{107-42\sqrt{2}}{-89} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-107+42\sqrt{2}}{89}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- 7 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3- 7 \sqrt{2}\right) } \cdot \left( 3- 7 \sqrt{2}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 7 \sqrt{2}\color{blue}{- 7 \sqrt{2}} \cdot3\color{blue}{- 7 \sqrt{2}} \cdot- 7 \sqrt{2} = \\ = 9- 21 \sqrt{2}- 21 \sqrt{2} + 98 $$ Simplify denominator. $$ \color{blue}{ \left( 3 + 7 \sqrt{2}\right) } \cdot \left( 3- 7 \sqrt{2}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 7 \sqrt{2}+\color{blue}{ 7 \sqrt{2}} \cdot3+\color{blue}{ 7 \sqrt{2}} \cdot- 7 \sqrt{2} = \\ = 9- 21 \sqrt{2} + 21 \sqrt{2}-98 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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