Question
Answer
$$\dfrac{1}{11+\sqrt{10}}=\dfrac{11-\sqrt{10}}{111}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{11+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{11+\sqrt{10}}\frac{11-\sqrt{10}}{11-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11-\sqrt{10}}{121-11\sqrt{10}+11\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{11-\sqrt{10}}{111}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 11- \sqrt{10}\right) = \color{blue}{1} \cdot11+\color{blue}{1} \cdot- \sqrt{10} = \\ = 11- \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 11 + \sqrt{10}\right) } \cdot \left( 11- \sqrt{10}\right) = \color{blue}{11} \cdot11+\color{blue}{11} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot11+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 121- 11 \sqrt{10} + 11 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
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