Question
Answer
$$\dfrac{3\sqrt{5}}{1+4\sqrt{3}}=\dfrac{-3\sqrt{5}+12\sqrt{15}}{47}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{5}}{1+4\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{5}}{1+4\sqrt{3}}\frac{1-4\sqrt{3}}{1-4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{5}-12\sqrt{15}}{1-4\sqrt{3}+4\sqrt{3}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3\sqrt{5}-12\sqrt{15}}{-47} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3\sqrt{5}+12\sqrt{15}}{47}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1- 4 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{5} } \cdot \left( 1- 4 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{5}} \cdot1+\color{blue}{ 3 \sqrt{5}} \cdot- 4 \sqrt{3} = \\ = 3 \sqrt{5}- 12 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 1 + 4 \sqrt{3}\right) } \cdot \left( 1- 4 \sqrt{3}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot1+\color{blue}{ 4 \sqrt{3}} \cdot- 4 \sqrt{3} = \\ = 1- 4 \sqrt{3} + 4 \sqrt{3}-48 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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