Answer
$$\frac{\sqrt{5}+2\sqrt{2}}{4-\sqrt{5}}=\frac{4\sqrt{5}+5+8\sqrt{2}+2\sqrt{10}}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+2\sqrt{2}}{4-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+2\sqrt{2}}{4-\sqrt{5}}\frac{4+\sqrt{5}}{4+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{5}+5+8\sqrt{2}+2\sqrt{10}}{16+4\sqrt{5}-4\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{5}+5+8\sqrt{2}+2\sqrt{10}}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + 2 \sqrt{2}\right) } \cdot \left( 4 + \sqrt{5}\right) = \color{blue}{ \sqrt{5}} \cdot4+\color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ 2 \sqrt{2}} \cdot4+\color{blue}{ 2 \sqrt{2}} \cdot \sqrt{5} = \\ = 4 \sqrt{5} + 5 + 8 \sqrt{2} + 2 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{5}\right) } \cdot \left( 4 + \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot4\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 16 + 4 \sqrt{5}- 4 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator