Question
Answer
$$\dfrac{\sqrt{5}-\sqrt{7}}{\sqrt{5}+\sqrt{7}}=-6+\sqrt{35}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}+\sqrt{7}}\frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5-\sqrt{35}-\sqrt{35}+7}{5-\sqrt{35}+\sqrt{35}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12-2\sqrt{35}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6-\sqrt{35}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-6+\sqrt{35}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-6+\sqrt{35}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5}- \sqrt{7}\right) } \cdot \left( \sqrt{5}- \sqrt{7}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{7}\color{blue}{- \sqrt{7}} \cdot \sqrt{5}\color{blue}{- \sqrt{7}} \cdot- \sqrt{7} = \\ = 5- \sqrt{35}- \sqrt{35} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{7}\right) } \cdot \left( \sqrt{5}- \sqrt{7}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot \sqrt{5}+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 5- \sqrt{35} + \sqrt{35}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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