Question
Answer
$$\dfrac{\sqrt{4}}{\sqrt{9}-\sqrt{2}}=\dfrac{6+2\sqrt{2}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{4}}{\sqrt{9}-\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{4}}{\sqrt{9}-\sqrt{2}}\frac{\sqrt{9}+\sqrt{2}}{\sqrt{9}+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+2\sqrt{2}}{9+3\sqrt{2}-3\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6+2\sqrt{2}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{9} + \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{4} } \cdot \left( \sqrt{9} + \sqrt{2}\right) = \color{blue}{ \sqrt{4}} \cdot \sqrt{9}+\color{blue}{ \sqrt{4}} \cdot \sqrt{2} = \\ = 6 + 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{9}- \sqrt{2}\right) } \cdot \left( \sqrt{9} + \sqrt{2}\right) = \color{blue}{ \sqrt{9}} \cdot \sqrt{9}+\color{blue}{ \sqrt{9}} \cdot \sqrt{2}\color{blue}{- \sqrt{2}} \cdot \sqrt{9}\color{blue}{- \sqrt{2}} \cdot \sqrt{2} = \\ = 9 + 3 \sqrt{2}- 3 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
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