Question
Answer
$$\dfrac{2\sqrt{3}}{3\sqrt{5}-4\sqrt{3}}=-\dfrac{6\sqrt{15}+24}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{3}}{3\sqrt{5}-4\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{3}}{3\sqrt{5}-4\sqrt{3}}\frac{3\sqrt{5}+4\sqrt{3}}{3\sqrt{5}+4\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{15}+24}{45+12\sqrt{15}-12\sqrt{15}-48} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{15}+24}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{6\sqrt{15}+24}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 4 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 \sqrt{3} } \cdot \left( 3 \sqrt{5} + 4 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 3 \sqrt{5}+\color{blue}{ 2 \sqrt{3}} \cdot 4 \sqrt{3} = \\ = 6 \sqrt{15} + 24 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}- 4 \sqrt{3}\right) } \cdot \left( 3 \sqrt{5} + 4 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot 4 \sqrt{3}\color{blue}{- 4 \sqrt{3}} \cdot 3 \sqrt{5}\color{blue}{- 4 \sqrt{3}} \cdot 4 \sqrt{3} = \\ = 45 + 12 \sqrt{15}- 12 \sqrt{15}-48 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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