Question
Answer
$$\dfrac{4\sqrt{2}}{3-\sqrt{5}}=3\sqrt{2}+\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{2}}{3-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{2}}{3-\sqrt{5}}\frac{3+\sqrt{5}}{3+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12\sqrt{2}+4\sqrt{10}}{9+3\sqrt{5}-3\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{2}+4\sqrt{10}}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{2}+\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3\sqrt{2}+\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 \sqrt{2} } \cdot \left( 3 + \sqrt{5}\right) = \color{blue}{ 4 \sqrt{2}} \cdot3+\color{blue}{ 4 \sqrt{2}} \cdot \sqrt{5} = \\ = 12 \sqrt{2} + 4 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 3- \sqrt{5}\right) } \cdot \left( 3 + \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot3\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 9 + 3 \sqrt{5}- 3 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator