Answer
$$\frac{\sqrt{2}}{\sqrt{10}+\sqrt{8}}=\sqrt{5}-2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{\sqrt{10}+\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{\sqrt{10}+\sqrt{8}}\frac{\sqrt{10}-\sqrt{8}}{\sqrt{10}-\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{5}-4}{10-4\sqrt{5}+4\sqrt{5}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{5}-4}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}-2}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\sqrt{5}-2\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10}- \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( \sqrt{10}- \sqrt{8}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{10}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{8} = \\ = 2 \sqrt{5}-4 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10} + \sqrt{8}\right) } \cdot \left( \sqrt{10}- \sqrt{8}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot- \sqrt{8}+\color{blue}{ \sqrt{8}} \cdot \sqrt{10}+\color{blue}{ \sqrt{8}} \cdot- \sqrt{8} = \\ = 10- 4 \sqrt{5} + 4 \sqrt{5}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator