Answer
$$\frac{\sqrt{2}}{9-\sqrt{3}}=\frac{9\sqrt{2}+\sqrt{6}}{78}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{9-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{9-\sqrt{3}}\frac{9+\sqrt{3}}{9+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{2}+\sqrt{6}}{81+9\sqrt{3}-9\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{2}+\sqrt{6}}{78}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9 + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( 9 + \sqrt{3}\right) = \color{blue}{ \sqrt{2}} \cdot9+\color{blue}{ \sqrt{2}} \cdot \sqrt{3} = \\ = 9 \sqrt{2} + \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 9- \sqrt{3}\right) } \cdot \left( 9 + \sqrt{3}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot9\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 81 + 9 \sqrt{3}- 9 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
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