Answer
$$\frac{4+\sqrt{27}}{2-3\sqrt{27}}=-\frac{89+42\sqrt{3}}{239}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+\sqrt{27}}{2-3\sqrt{27}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+\sqrt{27}}{2-3\sqrt{27}}\frac{2+3\sqrt{27}}{2+3\sqrt{27}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8+36\sqrt{3}+6\sqrt{3}+81}{4+18\sqrt{3}-18\sqrt{3}-243} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{89+42\sqrt{3}}{-239} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{89+42\sqrt{3}}{239}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 3 \sqrt{27}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + \sqrt{27}\right) } \cdot \left( 2 + 3 \sqrt{27}\right) = \color{blue}{4} \cdot2+\color{blue}{4} \cdot 3 \sqrt{27}+\color{blue}{ \sqrt{27}} \cdot2+\color{blue}{ \sqrt{27}} \cdot 3 \sqrt{27} = \\ = 8 + 36 \sqrt{3} + 6 \sqrt{3} + 81 $$ Simplify denominator. $$ \color{blue}{ \left( 2- 3 \sqrt{27}\right) } \cdot \left( 2 + 3 \sqrt{27}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 3 \sqrt{27}\color{blue}{- 3 \sqrt{27}} \cdot2\color{blue}{- 3 \sqrt{27}} \cdot 3 \sqrt{27} = \\ = 4 + 18 \sqrt{3}- 18 \sqrt{3}-243 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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