Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{\sqrt{7}}{2\sqrt{5}-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}}{2\sqrt{5}-\sqrt{7}}\frac{2\sqrt{5}+\sqrt{7}}{2\sqrt{5}+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{35}+7}{20+2\sqrt{35}-2\sqrt{35}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{35}+7}{13}\end{aligned} $$ | |
① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5} + \sqrt{7}} $$. |
② | Multiply in a numerator. $$ \color{blue}{ \sqrt{7} } \cdot \left( 2 \sqrt{5} + \sqrt{7}\right) = \color{blue}{ \sqrt{7}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{7}} \cdot \sqrt{7} = \\ = 2 \sqrt{35} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5}- \sqrt{7}\right) } \cdot \left( 2 \sqrt{5} + \sqrt{7}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot 2 \sqrt{5}\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 20 + 2 \sqrt{35}- 2 \sqrt{35}-7 $$ |
③ | Simplify numerator and denominator |