Answer
$$\frac{4}{3\sqrt{3}-2}=\frac{12\sqrt{3}+8}{23}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{3\sqrt{3}-2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{3\sqrt{3}-2}\frac{3\sqrt{3}+2}{3\sqrt{3}+2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12\sqrt{3}+8}{27+6\sqrt{3}-6\sqrt{3}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{3}+8}{23}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{3} + 2} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 3 \sqrt{3} + 2\right) = \color{blue}{4} \cdot 3 \sqrt{3}+\color{blue}{4} \cdot2 = \\ = 12 \sqrt{3} + 8 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{3}-2\right) } \cdot \left( 3 \sqrt{3} + 2\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot2\color{blue}{-2} \cdot 3 \sqrt{3}\color{blue}{-2} \cdot2 = \\ = 27 + 6 \sqrt{3}- 6 \sqrt{3}-4 $$ |
| ③ | Simplify numerator and denominator |
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