Question
Answer
$$\dfrac{\sqrt{5}+\sqrt{12}}{\sqrt{6}+\sqrt{10}}=\dfrac{\sqrt{30}-\sqrt{2}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+\sqrt{12}}{\sqrt{6}+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+\sqrt{12}}{\sqrt{6}+\sqrt{10}}\frac{\sqrt{6}-\sqrt{10}}{\sqrt{6}-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{30}-5\sqrt{2}+6\sqrt{2}-2\sqrt{30}}{6-2\sqrt{15}+2\sqrt{15}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-\sqrt{30}+\sqrt{2}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{30}-\sqrt{2}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{6}- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{12}\right) } \cdot \left( \sqrt{6}- \sqrt{10}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{6}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{10}+\color{blue}{ \sqrt{12}} \cdot \sqrt{6}+\color{blue}{ \sqrt{12}} \cdot- \sqrt{10} = \\ = \sqrt{30}- 5 \sqrt{2} + 6 \sqrt{2}- 2 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6} + \sqrt{10}\right) } \cdot \left( \sqrt{6}- \sqrt{10}\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot \sqrt{6}+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 6- 2 \sqrt{15} + 2 \sqrt{15}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator