Answer
$$\frac{1+\sqrt{5}}{1+\sqrt{7}}=\frac{-1+\sqrt{7}-\sqrt{5}+\sqrt{35}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+\sqrt{5}}{1+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1+\sqrt{5}}{1+\sqrt{7}}\frac{1-\sqrt{7}}{1-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1-\sqrt{7}+\sqrt{5}-\sqrt{35}}{1-\sqrt{7}+\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1-\sqrt{7}+\sqrt{5}-\sqrt{35}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-1+\sqrt{7}-\sqrt{5}+\sqrt{35}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1 + \sqrt{5}\right) } \cdot \left( 1- \sqrt{7}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- \sqrt{7}+\color{blue}{ \sqrt{5}} \cdot1+\color{blue}{ \sqrt{5}} \cdot- \sqrt{7} = \\ = 1- \sqrt{7} + \sqrt{5}- \sqrt{35} $$ Simplify denominator. $$ \color{blue}{ \left( 1 + \sqrt{7}\right) } \cdot \left( 1- \sqrt{7}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot1+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 1- \sqrt{7} + \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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