Question
Answer
$$\dfrac{5-3\sqrt{2}}{6+5\sqrt{2}}=\dfrac{-60+43\sqrt{2}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5-3\sqrt{2}}{6+5\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5-3\sqrt{2}}{6+5\sqrt{2}}\frac{6-5\sqrt{2}}{6-5\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30-25\sqrt{2}-18\sqrt{2}+30}{36-30\sqrt{2}+30\sqrt{2}-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60-43\sqrt{2}}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-60+43\sqrt{2}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 5 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5- 3 \sqrt{2}\right) } \cdot \left( 6- 5 \sqrt{2}\right) = \color{blue}{5} \cdot6+\color{blue}{5} \cdot- 5 \sqrt{2}\color{blue}{- 3 \sqrt{2}} \cdot6\color{blue}{- 3 \sqrt{2}} \cdot- 5 \sqrt{2} = \\ = 30- 25 \sqrt{2}- 18 \sqrt{2} + 30 $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 5 \sqrt{2}\right) } \cdot \left( 6- 5 \sqrt{2}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot6+\color{blue}{ 5 \sqrt{2}} \cdot- 5 \sqrt{2} = \\ = 36- 30 \sqrt{2} + 30 \sqrt{2}-50 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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