Question
Answer
$$\dfrac{4+\sqrt{2}}{5-\sqrt{6}}=\dfrac{20+4\sqrt{6}+5\sqrt{2}+2\sqrt{3}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+\sqrt{2}}{5-\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+\sqrt{2}}{5-\sqrt{6}}\frac{5+\sqrt{6}}{5+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20+4\sqrt{6}+5\sqrt{2}+2\sqrt{3}}{25+5\sqrt{6}-5\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20+4\sqrt{6}+5\sqrt{2}+2\sqrt{3}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + \sqrt{2}\right) } \cdot \left( 5 + \sqrt{6}\right) = \color{blue}{4} \cdot5+\color{blue}{4} \cdot \sqrt{6}+\color{blue}{ \sqrt{2}} \cdot5+\color{blue}{ \sqrt{2}} \cdot \sqrt{6} = \\ = 20 + 4 \sqrt{6} + 5 \sqrt{2} + 2 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 5- \sqrt{6}\right) } \cdot \left( 5 + \sqrt{6}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{6}\color{blue}{- \sqrt{6}} \cdot5\color{blue}{- \sqrt{6}} \cdot \sqrt{6} = \\ = 25 + 5 \sqrt{6}- 5 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator