Question
Answer
$$\dfrac{\dfrac{2^3}{2}}{\sqrt{3}+6}=\dfrac{-4\sqrt{3}+24}{33}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{2^3}{2}}{\sqrt{3}+6}& \xlongequal{ }\frac{\frac{8}{2}}{\sqrt{3}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{ \frac{ 8 : \color{orangered}{ 2 } }{ 2 : \color{orangered}{ 2 }} }{ \sqrt{3}+6 } \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{\frac{4}{1}}{\sqrt{3}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4}{\sqrt{3}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4}{\sqrt{3}+6}\frac{\sqrt{3}-6}{\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{3}-24}{3-6\sqrt{3}+6\sqrt{3}-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4\sqrt{3}-24}{-33} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{-4\sqrt{3}+24}{33}\end{aligned} $$ | |
| ① | Divide both the top and bottom numbers by $ \color{orangered}{ 2 } $. |
| ② | Remove 1 from denominator. |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}-6} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{3}-6\right) = \color{blue}{4} \cdot \sqrt{3}+\color{blue}{4} \cdot-6 = \\ = 4 \sqrt{3}-24 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + 6\right) } \cdot \left( \sqrt{3}-6\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot-6+\color{blue}{6} \cdot \sqrt{3}+\color{blue}{6} \cdot-6 = \\ = 3- 6 \sqrt{3} + 6 \sqrt{3}-36 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Multiply both numerator and denominator by -1. |
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