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