Answer
$$\frac{4}{\sqrt{2}-5\sqrt{3}}=-\frac{4\sqrt{2}+20\sqrt{3}}{73}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{2}-5\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{2}-5\sqrt{3}}\frac{\sqrt{2}+5\sqrt{3}}{\sqrt{2}+5\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{2}+20\sqrt{3}}{2+5\sqrt{6}-5\sqrt{6}-75} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{2}+20\sqrt{3}}{-73} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{4\sqrt{2}+20\sqrt{3}}{73}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2} + 5 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{2} + 5 \sqrt{3}\right) = \color{blue}{4} \cdot \sqrt{2}+\color{blue}{4} \cdot 5 \sqrt{3} = \\ = 4 \sqrt{2} + 20 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2}- 5 \sqrt{3}\right) } \cdot \left( \sqrt{2} + 5 \sqrt{3}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 5 \sqrt{3}\color{blue}{- 5 \sqrt{3}} \cdot \sqrt{2}\color{blue}{- 5 \sqrt{3}} \cdot 5 \sqrt{3} = \\ = 2 + 5 \sqrt{6}- 5 \sqrt{6}-75 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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