Question
Answer
$$\dfrac{3i^{30}-i^{19}}{-1+2i}=i+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3i^{30}-i^{19}}{-1+2i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-3+i}{-1+2i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}i+1\end{aligned} $$ | |
| ① | $$ 3i^{30} = 3 \cdot i^{4 \cdot 7 + 2} =
3 \cdot \left( i^4 \right)^{ 7 } \cdot i^2 =
3 \cdot 1^{ 7 } \cdot (-1) =
3 \cdot -1 = -3 $$$$ -i^{19} = - i^{4 \cdot 4 + 3} =
- \left( i^4 \right)^{ 4 } \cdot i^3 =
- 1^{ 4 } \cdot (-i) =
- -i = i $$ |
| ② | Divide $ \, -3+i \, $ by $ \, -1+2i \, $ to get $\,\, 1+i $. ( view steps ) |
| ③ | Combine like terms: $$ i+1 = i+1 $$ |
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