Question
Answer
$$\dfrac{1}{\sqrt{7}+4\sqrt{3}}=\dfrac{-\sqrt{7}+4\sqrt{3}}{41}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\sqrt{7}+4\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{\sqrt{7}+4\sqrt{3}}\frac{\sqrt{7}-4\sqrt{3}}{\sqrt{7}-4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{7}-4\sqrt{3}}{7-4\sqrt{21}+4\sqrt{21}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{7}-4\sqrt{3}}{-41} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{7}+4\sqrt{3}}{41}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7}- 4 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( \sqrt{7}- 4 \sqrt{3}\right) = \color{blue}{1} \cdot \sqrt{7}+\color{blue}{1} \cdot- 4 \sqrt{3} = \\ = \sqrt{7}- 4 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7} + 4 \sqrt{3}\right) } \cdot \left( \sqrt{7}- 4 \sqrt{3}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot- 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot \sqrt{7}+\color{blue}{ 4 \sqrt{3}} \cdot- 4 \sqrt{3} = \\ = 7- 4 \sqrt{21} + 4 \sqrt{21}-48 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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