Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{\sqrt{41}+\sqrt{29}}{\sqrt{41}-\sqrt{29}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{41}+\sqrt{29}}{\sqrt{41}-\sqrt{29}}\frac{\sqrt{41}+\sqrt{29}}{\sqrt{41}+\sqrt{29}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{41+\sqrt{1189}+\sqrt{1189}+29}{41+\sqrt{1189}-\sqrt{1189}-29} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{70+2\sqrt{1189}}{12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{35+\sqrt{1189}}{6}\end{aligned} $$ | |
① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{41} + \sqrt{29}} $$. |
② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{41} + \sqrt{29}\right) } \cdot \left( \sqrt{41} + \sqrt{29}\right) = \color{blue}{ \sqrt{41}} \cdot \sqrt{41}+\color{blue}{ \sqrt{41}} \cdot \sqrt{29}+\color{blue}{ \sqrt{29}} \cdot \sqrt{41}+\color{blue}{ \sqrt{29}} \cdot \sqrt{29} = \\ = 41 + \sqrt{1189} + \sqrt{1189} + 29 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{41}- \sqrt{29}\right) } \cdot \left( \sqrt{41} + \sqrt{29}\right) = \color{blue}{ \sqrt{41}} \cdot \sqrt{41}+\color{blue}{ \sqrt{41}} \cdot \sqrt{29}\color{blue}{- \sqrt{29}} \cdot \sqrt{41}\color{blue}{- \sqrt{29}} \cdot \sqrt{29} = \\ = 41 + \sqrt{1189}- \sqrt{1189}-29 $$ |
③ | Simplify numerator and denominator |
④ | Divide both numerator and denominator by 2. |