Answer
$$\frac{4}{8-3\sqrt{5}}=\frac{32+12\sqrt{5}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{8-3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{8-3\sqrt{5}}\frac{8+3\sqrt{5}}{8+3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{32+12\sqrt{5}}{64+24\sqrt{5}-24\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{32+12\sqrt{5}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8 + 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 8 + 3 \sqrt{5}\right) = \color{blue}{4} \cdot8+\color{blue}{4} \cdot 3 \sqrt{5} = \\ = 32 + 12 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 8- 3 \sqrt{5}\right) } \cdot \left( 8 + 3 \sqrt{5}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot 3 \sqrt{5}\color{blue}{- 3 \sqrt{5}} \cdot8\color{blue}{- 3 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 64 + 24 \sqrt{5}- 24 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
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