Question
Answer
$$\dfrac{\sqrt{5}+\sqrt{2}}{9+2\sqrt{14}}=\dfrac{9\sqrt{5}-2\sqrt{70}+9\sqrt{2}-4\sqrt{7}}{25}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+\sqrt{2}}{9+2\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+\sqrt{2}}{9+2\sqrt{14}}\frac{9-2\sqrt{14}}{9-2\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{5}-2\sqrt{70}+9\sqrt{2}-4\sqrt{7}}{81-18\sqrt{14}+18\sqrt{14}-56} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{5}-2\sqrt{70}+9\sqrt{2}-4\sqrt{7}}{25}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9- 2 \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{2}\right) } \cdot \left( 9- 2 \sqrt{14}\right) = \color{blue}{ \sqrt{5}} \cdot9+\color{blue}{ \sqrt{5}} \cdot- 2 \sqrt{14}+\color{blue}{ \sqrt{2}} \cdot9+\color{blue}{ \sqrt{2}} \cdot- 2 \sqrt{14} = \\ = 9 \sqrt{5}- 2 \sqrt{70} + 9 \sqrt{2}- 4 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 9 + 2 \sqrt{14}\right) } \cdot \left( 9- 2 \sqrt{14}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- 2 \sqrt{14}+\color{blue}{ 2 \sqrt{14}} \cdot9+\color{blue}{ 2 \sqrt{14}} \cdot- 2 \sqrt{14} = \\ = 81- 18 \sqrt{14} + 18 \sqrt{14}-56 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator