Answer
$$\frac{\sqrt{2}}{2\sqrt{5}-3\sqrt{2}}=\sqrt{10}+3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{2\sqrt{5}-3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{2\sqrt{5}-3\sqrt{2}}\frac{2\sqrt{5}+3\sqrt{2}}{2\sqrt{5}+3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{10}+6}{20+6\sqrt{10}-6\sqrt{10}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{10}+6}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{10}+3}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\sqrt{10}+3\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5} + 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( 2 \sqrt{5} + 3 \sqrt{2}\right) = \color{blue}{ \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 2 \sqrt{10} + 6 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5}- 3 \sqrt{2}\right) } \cdot \left( 2 \sqrt{5} + 3 \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot 3 \sqrt{2}\color{blue}{- 3 \sqrt{2}} \cdot 2 \sqrt{5}\color{blue}{- 3 \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 20 + 6 \sqrt{10}- 6 \sqrt{10}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator