Answer
$$\frac{2+3\sqrt{7}}{5-2\sqrt{7}}=-\frac{52+19\sqrt{7}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+3\sqrt{7}}{5-2\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+3\sqrt{7}}{5-2\sqrt{7}}\frac{5+2\sqrt{7}}{5+2\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+4\sqrt{7}+15\sqrt{7}+42}{25+10\sqrt{7}-10\sqrt{7}-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{52+19\sqrt{7}}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{52+19\sqrt{7}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 2 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + 3 \sqrt{7}\right) } \cdot \left( 5 + 2 \sqrt{7}\right) = \color{blue}{2} \cdot5+\color{blue}{2} \cdot 2 \sqrt{7}+\color{blue}{ 3 \sqrt{7}} \cdot5+\color{blue}{ 3 \sqrt{7}} \cdot 2 \sqrt{7} = \\ = 10 + 4 \sqrt{7} + 15 \sqrt{7} + 42 $$ Simplify denominator. $$ \color{blue}{ \left( 5- 2 \sqrt{7}\right) } \cdot \left( 5 + 2 \sqrt{7}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 2 \sqrt{7}\color{blue}{- 2 \sqrt{7}} \cdot5\color{blue}{- 2 \sqrt{7}} \cdot 2 \sqrt{7} = \\ = 25 + 10 \sqrt{7}- 10 \sqrt{7}-28 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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