Answer
$$\frac{\sqrt{35}}{9+\sqrt{10}}=\frac{9\sqrt{35}-5\sqrt{14}}{71}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{35}}{9+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{35}}{9+\sqrt{10}}\frac{9-\sqrt{10}}{9-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{35}-5\sqrt{14}}{81-9\sqrt{10}+9\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{35}-5\sqrt{14}}{71}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{35} } \cdot \left( 9- \sqrt{10}\right) = \color{blue}{ \sqrt{35}} \cdot9+\color{blue}{ \sqrt{35}} \cdot- \sqrt{10} = \\ = 9 \sqrt{35}- 5 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 9 + \sqrt{10}\right) } \cdot \left( 9- \sqrt{10}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot9+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 81- 9 \sqrt{10} + 9 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
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