Answer
$$\frac{5}{2\sqrt{6}+3}=\frac{2\sqrt{6}-3}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{2\sqrt{6}+3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{2\sqrt{6}+3}\frac{2\sqrt{6}-3}{2\sqrt{6}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10\sqrt{6}-15}{24-6\sqrt{6}+6\sqrt{6}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{10\sqrt{6}-15}{15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{6}-3}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{6}-3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 2 \sqrt{6}-3\right) = \color{blue}{5} \cdot 2 \sqrt{6}+\color{blue}{5} \cdot-3 = \\ = 10 \sqrt{6}-15 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{6} + 3\right) } \cdot \left( 2 \sqrt{6}-3\right) = \color{blue}{ 2 \sqrt{6}} \cdot 2 \sqrt{6}+\color{blue}{ 2 \sqrt{6}} \cdot-3+\color{blue}{3} \cdot 2 \sqrt{6}+\color{blue}{3} \cdot-3 = \\ = 24- 6 \sqrt{6} + 6 \sqrt{6}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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