Question
Answer
$$\dfrac{11+\sqrt{41}}{21-\sqrt{41}}=\dfrac{17+2\sqrt{41}}{25}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{11+\sqrt{41}}{21-\sqrt{41}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{11+\sqrt{41}}{21-\sqrt{41}}\frac{21+\sqrt{41}}{21+\sqrt{41}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{231+11\sqrt{41}+21\sqrt{41}+41}{441+21\sqrt{41}-21\sqrt{41}-41} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{272+32\sqrt{41}}{400} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{17+2\sqrt{41}}{25}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 21 + \sqrt{41}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 11 + \sqrt{41}\right) } \cdot \left( 21 + \sqrt{41}\right) = \color{blue}{11} \cdot21+\color{blue}{11} \cdot \sqrt{41}+\color{blue}{ \sqrt{41}} \cdot21+\color{blue}{ \sqrt{41}} \cdot \sqrt{41} = \\ = 231 + 11 \sqrt{41} + 21 \sqrt{41} + 41 $$ Simplify denominator. $$ \color{blue}{ \left( 21- \sqrt{41}\right) } \cdot \left( 21 + \sqrt{41}\right) = \color{blue}{21} \cdot21+\color{blue}{21} \cdot \sqrt{41}\color{blue}{- \sqrt{41}} \cdot21\color{blue}{- \sqrt{41}} \cdot \sqrt{41} = \\ = 441 + 21 \sqrt{41}- 21 \sqrt{41}-41 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 16. |
This page was created using
Rationalize Denominator Calculator