Answer
$$\frac{3}{-6+9\sqrt{3}}=\frac{2+3\sqrt{3}}{23}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{-6+9\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{-6+9\sqrt{3}}\frac{-6-9\sqrt{3}}{-6-9\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-18-27\sqrt{3}}{36+54\sqrt{3}-54\sqrt{3}-243} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-18-27\sqrt{3}}{-207} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-2-3\sqrt{3}}{-23} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{2+3\sqrt{3}}{23}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -6- 9 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( -6- 9 \sqrt{3}\right) = \color{blue}{3} \cdot-6+\color{blue}{3} \cdot- 9 \sqrt{3} = \\ = -18- 27 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( -6 + 9 \sqrt{3}\right) } \cdot \left( -6- 9 \sqrt{3}\right) = \color{blue}{-6} \cdot-6\color{blue}{-6} \cdot- 9 \sqrt{3}+\color{blue}{ 9 \sqrt{3}} \cdot-6+\color{blue}{ 9 \sqrt{3}} \cdot- 9 \sqrt{3} = \\ = 36 + 54 \sqrt{3}- 54 \sqrt{3}-243 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator