Question
Answer
$$\dfrac{4}{\sqrt{3}+6\sqrt{10}}=\dfrac{-4\sqrt{3}+24\sqrt{10}}{357}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{3}+6\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{3}+6\sqrt{10}}\frac{\sqrt{3}-6\sqrt{10}}{\sqrt{3}-6\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{3}-24\sqrt{10}}{3-6\sqrt{30}+6\sqrt{30}-360} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{3}-24\sqrt{10}}{-357} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-4\sqrt{3}+24\sqrt{10}}{357}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}- 6 \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{3}- 6 \sqrt{10}\right) = \color{blue}{4} \cdot \sqrt{3}+\color{blue}{4} \cdot- 6 \sqrt{10} = \\ = 4 \sqrt{3}- 24 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + 6 \sqrt{10}\right) } \cdot \left( \sqrt{3}- 6 \sqrt{10}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot- 6 \sqrt{10}+\color{blue}{ 6 \sqrt{10}} \cdot \sqrt{3}+\color{blue}{ 6 \sqrt{10}} \cdot- 6 \sqrt{10} = \\ = 3- 6 \sqrt{30} + 6 \sqrt{30}-360 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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