Answer
$$\frac{5+2\sqrt{3}}{2+\sqrt{3}}=4-\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5+2\sqrt{3}}{2+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5+2\sqrt{3}}{2+\sqrt{3}}\frac{2-\sqrt{3}}{2-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10-5\sqrt{3}+4\sqrt{3}-6}{4-2\sqrt{3}+2\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4-\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4-\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 + 2 \sqrt{3}\right) } \cdot \left( 2- \sqrt{3}\right) = \color{blue}{5} \cdot2+\color{blue}{5} \cdot- \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot2+\color{blue}{ 2 \sqrt{3}} \cdot- \sqrt{3} = \\ = 10- 5 \sqrt{3} + 4 \sqrt{3}-6 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{3}\right) } \cdot \left( 2- \sqrt{3}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot2+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 4- 2 \sqrt{3} + 2 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Remove 1 from denominator. |
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