Question
Answer
$$\dfrac{\sqrt{2}}{\sqrt{5}-\sqrt{7}}=-\dfrac{\sqrt{10}+\sqrt{14}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{\sqrt{5}-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{\sqrt{5}-\sqrt{7}}\frac{\sqrt{5}+\sqrt{7}}{\sqrt{5}+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{10}+\sqrt{14}}{5+\sqrt{35}-\sqrt{35}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{10}+\sqrt{14}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{\sqrt{10}+\sqrt{14}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( \sqrt{5} + \sqrt{7}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot \sqrt{7} = \\ = \sqrt{10} + \sqrt{14} $$ 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 | Place a negative sign in front of a fraction. |
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