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