Question
Answer
$$\dfrac{5}{\sqrt{2}+\sqrt{7}}=-\sqrt{2}+\sqrt{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{\sqrt{2}+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{\sqrt{2}+\sqrt{7}}\frac{\sqrt{2}-\sqrt{7}}{\sqrt{2}-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{2}-5\sqrt{7}}{2-\sqrt{14}+\sqrt{14}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{2}-5\sqrt{7}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}-\sqrt{7}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-\sqrt{2}+\sqrt{7}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-\sqrt{2}+\sqrt{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2}- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( \sqrt{2}- \sqrt{7}\right) = \color{blue}{5} \cdot \sqrt{2}+\color{blue}{5} \cdot- \sqrt{7} = \\ = 5 \sqrt{2}- 5 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2} + \sqrt{7}\right) } \cdot \left( \sqrt{2}- \sqrt{7}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot \sqrt{2}+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 2- \sqrt{14} + \sqrt{14}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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