Answer
$$\frac{2+3\sqrt{3}}{3-4\sqrt{5}}=-\frac{6+8\sqrt{5}+9\sqrt{3}+12\sqrt{15}}{71}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+3\sqrt{3}}{3-4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+3\sqrt{3}}{3-4\sqrt{5}}\frac{3+4\sqrt{5}}{3+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+8\sqrt{5}+9\sqrt{3}+12\sqrt{15}}{9+12\sqrt{5}-12\sqrt{5}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6+8\sqrt{5}+9\sqrt{3}+12\sqrt{15}}{-71} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{6+8\sqrt{5}+9\sqrt{3}+12\sqrt{15}}{71}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + 3 \sqrt{3}\right) } \cdot \left( 3 + 4 \sqrt{5}\right) = \color{blue}{2} \cdot3+\color{blue}{2} \cdot 4 \sqrt{5}+\color{blue}{ 3 \sqrt{3}} \cdot3+\color{blue}{ 3 \sqrt{3}} \cdot 4 \sqrt{5} = \\ = 6 + 8 \sqrt{5} + 9 \sqrt{3} + 12 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 3- 4 \sqrt{5}\right) } \cdot \left( 3 + 4 \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot3\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 9 + 12 \sqrt{5}- 12 \sqrt{5}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator