Answer
$$\frac{2+\sqrt{7}}{5+\sqrt{7}}=\frac{1+\sqrt{7}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+\sqrt{7}}{5+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+\sqrt{7}}{5+\sqrt{7}}\frac{5-\sqrt{7}}{5-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10-2\sqrt{7}+5\sqrt{7}-7}{25-5\sqrt{7}+5\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3+3\sqrt{7}}{18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\sqrt{7}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + \sqrt{7}\right) } \cdot \left( 5- \sqrt{7}\right) = \color{blue}{2} \cdot5+\color{blue}{2} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot5+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 10- 2 \sqrt{7} + 5 \sqrt{7}-7 $$ Simplify denominator. $$ \color{blue}{ \left( 5 + \sqrt{7}\right) } \cdot \left( 5- \sqrt{7}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot5+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 25- 5 \sqrt{7} + 5 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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