Answer
$$\frac{2}{8+3\sqrt{5}}=\frac{16-6\sqrt{5}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{8+3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{8+3\sqrt{5}}\frac{8-3\sqrt{5}}{8-3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16-6\sqrt{5}}{64-24\sqrt{5}+24\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16-6\sqrt{5}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8- 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 8- 3 \sqrt{5}\right) = \color{blue}{2} \cdot8+\color{blue}{2} \cdot- 3 \sqrt{5} = \\ = 16- 6 \sqrt{5} $$ 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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