Question
Answer
$$\dfrac{\sqrt{2}-\sqrt{5}}{\sqrt{8}+\sqrt{5}}=3-\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}-\sqrt{5}}{\sqrt{8}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}-\sqrt{5}}{\sqrt{8}+\sqrt{5}}\frac{\sqrt{8}-\sqrt{5}}{\sqrt{8}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4-\sqrt{10}-2\sqrt{10}+5}{8-2\sqrt{10}+2\sqrt{10}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9-3\sqrt{10}}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3-\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{8}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2}- \sqrt{5}\right) } \cdot \left( \sqrt{8}- \sqrt{5}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{8}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{5}\color{blue}{- \sqrt{5}} \cdot \sqrt{8}\color{blue}{- \sqrt{5}} \cdot- \sqrt{5} = \\ = 4- \sqrt{10}- 2 \sqrt{10} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{8} + \sqrt{5}\right) } \cdot \left( \sqrt{8}- \sqrt{5}\right) = \color{blue}{ \sqrt{8}} \cdot \sqrt{8}+\color{blue}{ \sqrt{8}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{8}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 8- 2 \sqrt{10} + 2 \sqrt{10}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator