Question
Answer
$$\dfrac{8-2\sqrt{15}}{8+2\sqrt{15}}=31-8\sqrt{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8-2\sqrt{15}}{8+2\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8-2\sqrt{15}}{8+2\sqrt{15}}\frac{8-2\sqrt{15}}{8-2\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{64-16\sqrt{15}-16\sqrt{15}+60}{64-16\sqrt{15}+16\sqrt{15}-60} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{124-32\sqrt{15}}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{31-8\sqrt{15}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}31-8\sqrt{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8- 2 \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 8- 2 \sqrt{15}\right) } \cdot \left( 8- 2 \sqrt{15}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot- 2 \sqrt{15}\color{blue}{- 2 \sqrt{15}} \cdot8\color{blue}{- 2 \sqrt{15}} \cdot- 2 \sqrt{15} = \\ = 64- 16 \sqrt{15}- 16 \sqrt{15} + 60 $$ Simplify denominator. $$ \color{blue}{ \left( 8 + 2 \sqrt{15}\right) } \cdot \left( 8- 2 \sqrt{15}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot- 2 \sqrt{15}+\color{blue}{ 2 \sqrt{15}} \cdot8+\color{blue}{ 2 \sqrt{15}} \cdot- 2 \sqrt{15} = \\ = 64- 16 \sqrt{15} + 16 \sqrt{15}-60 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Remove 1 from denominator. |
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