Question
Answer
$$\dfrac{\sqrt{2}-5}{\sqrt{3}+6\sqrt{5}}=\dfrac{-\sqrt{6}+6\sqrt{10}+5\sqrt{3}-30\sqrt{5}}{177}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}-5}{\sqrt{3}+6\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}-5}{\sqrt{3}+6\sqrt{5}}\frac{\sqrt{3}-6\sqrt{5}}{\sqrt{3}-6\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{6}-6\sqrt{10}-5\sqrt{3}+30\sqrt{5}}{3-6\sqrt{15}+6\sqrt{15}-180} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{6}-6\sqrt{10}-5\sqrt{3}+30\sqrt{5}}{-177} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{6}+6\sqrt{10}+5\sqrt{3}-30\sqrt{5}}{177}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}- 6 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2}-5\right) } \cdot \left( \sqrt{3}- 6 \sqrt{5}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{3}+\color{blue}{ \sqrt{2}} \cdot- 6 \sqrt{5}\color{blue}{-5} \cdot \sqrt{3}\color{blue}{-5} \cdot- 6 \sqrt{5} = \\ = \sqrt{6}- 6 \sqrt{10}- 5 \sqrt{3} + 30 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + 6 \sqrt{5}\right) } \cdot \left( \sqrt{3}- 6 \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot- 6 \sqrt{5}+\color{blue}{ 6 \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ 6 \sqrt{5}} \cdot- 6 \sqrt{5} = \\ = 3- 6 \sqrt{15} + 6 \sqrt{15}-180 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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