Answer
$$\frac{7-\sqrt{5}}{7+\sqrt{5}}=\frac{27-7\sqrt{5}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7-\sqrt{5}}{7+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7-\sqrt{5}}{7+\sqrt{5}}\frac{7-\sqrt{5}}{7-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{49-7\sqrt{5}-7\sqrt{5}+5}{49-7\sqrt{5}+7\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{54-14\sqrt{5}}{44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{27-7\sqrt{5}}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7- \sqrt{5}\right) } \cdot \left( 7- \sqrt{5}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot- \sqrt{5}\color{blue}{- \sqrt{5}} \cdot7\color{blue}{- \sqrt{5}} \cdot- \sqrt{5} = \\ = 49- 7 \sqrt{5}- 7 \sqrt{5} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( 7 + \sqrt{5}\right) } \cdot \left( 7- \sqrt{5}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot7+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 49- 7 \sqrt{5} + 7 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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