Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{5+\sqrt{7}}{2-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5+\sqrt{7}}{2-\sqrt{5}}\frac{2+\sqrt{5}}{2+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}}{4+2\sqrt{5}-2\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-10-5\sqrt{5}-2\sqrt{7}-\sqrt{35}\end{aligned} $$ | |
① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + \sqrt{5}} $$. |
② | Multiply in a numerator. $$ \color{blue}{ \left( 5 + \sqrt{7}\right) } \cdot \left( 2 + \sqrt{5}\right) = \color{blue}{5} \cdot2+\color{blue}{5} \cdot \sqrt{5}+\color{blue}{ \sqrt{7}} \cdot2+\color{blue}{ \sqrt{7}} \cdot \sqrt{5} = \\ = 10 + 5 \sqrt{5} + 2 \sqrt{7} + \sqrt{35} $$ Simplify denominator. $$ \color{blue}{ \left( 2- \sqrt{5}\right) } \cdot \left( 2 + \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot2\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 4 + 2 \sqrt{5}- 2 \sqrt{5}-5 $$ |
③ | Simplify numerator and denominator |
④ | Place a negative sign in front of a fraction. |
⑤ | Remove the parenthesis by changing the sign of each term within them. |