Question
Answer
$$\dfrac{(3+\sqrt{7})(\sqrt{7}-3)}{\sqrt{5}+\sqrt{6}}=2\sqrt{5}-2\sqrt{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(3+\sqrt{7})(\sqrt{7}-3)}{\sqrt{5}+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{7}-9+7-3\sqrt{7}}{\sqrt{5}+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-2}{\sqrt{5}+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2}{\sqrt{5}+\sqrt{6}}\frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-2\sqrt{5}+2\sqrt{6}}{5-\sqrt{30}+\sqrt{30}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-2\sqrt{5}+2\sqrt{6}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{2\sqrt{5}-2\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}2\sqrt{5}-2\sqrt{6}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 3 + \sqrt{7}\right) } \cdot \left( \sqrt{7}-3\right) = \color{blue}{3} \cdot \sqrt{7}+\color{blue}{3} \cdot-3+\color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot-3 = \\ = 3 \sqrt{7}-9 + 7- 3 \sqrt{7} $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}- \sqrt{6}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ -2 } \cdot \left( \sqrt{5}- \sqrt{6}\right) = \color{blue}{-2} \cdot \sqrt{5}\color{blue}{-2} \cdot- \sqrt{6} = \\ = - 2 \sqrt{5} + 2 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{6}\right) } \cdot \left( \sqrt{5}- \sqrt{6}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot \sqrt{5}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 5- \sqrt{30} + \sqrt{30}-6 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Multiply both numerator and denominator by -1. |
| ⑦ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator