Question
Answer
$$\dfrac{\sqrt{3}}{\sqrt{5}+4\sqrt{10}}=\dfrac{-\sqrt{15}+4\sqrt{30}}{155}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}}{\sqrt{5}+4\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}}{\sqrt{5}+4\sqrt{10}}\frac{\sqrt{5}-4\sqrt{10}}{\sqrt{5}-4\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{15}-4\sqrt{30}}{5-20\sqrt{2}+20\sqrt{2}-160} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{15}-4\sqrt{30}}{-155} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{15}+4\sqrt{30}}{155}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}- 4 \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{3} } \cdot \left( \sqrt{5}- 4 \sqrt{10}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{5}+\color{blue}{ \sqrt{3}} \cdot- 4 \sqrt{10} = \\ = \sqrt{15}- 4 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5} + 4 \sqrt{10}\right) } \cdot \left( \sqrt{5}- 4 \sqrt{10}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot- 4 \sqrt{10}+\color{blue}{ 4 \sqrt{10}} \cdot \sqrt{5}+\color{blue}{ 4 \sqrt{10}} \cdot- 4 \sqrt{10} = \\ = 5- 20 \sqrt{2} + 20 \sqrt{2}-160 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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