Answer
$$5i^{19}-i^{112}+3i^{65}+7i^{106}=-2i-8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}5i^{19}-i^{112}+3i^{65}+7i^{106}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-5i-1+3i-7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-2i-8\end{aligned} $$ | |
| ① | $$ 5i^{19} = 5 \cdot i^{4 \cdot 4 + 3} =
5 \cdot \left( i^4 \right)^{ 4 } \cdot i^3 =
5 \cdot 1^{ 4 } \cdot (-i) =
5 \cdot -i = -5i $$ |
| ② | $$ -i^{112} = - i^{4 \cdot 28 + 0} =
- \left( i^4 \right)^{ 28 } \cdot i^0 =
- 1^{ 28 } \cdot 1 =
- 1 $$ |
| ③ | $$ 3i^{65} = 3 \cdot i^{4 \cdot 16 + 1} =
3 \cdot \left( i^4 \right)^{ 16 } \cdot i^1 =
3 \cdot 1^{ 16 } \cdot i =
3 \cdot i $$ |
| ④ | $$ 7i^{106} = 7 \cdot i^{4 \cdot 26 + 2} =
7 \cdot \left( i^4 \right)^{ 26 } \cdot i^2 =
7 \cdot 1^{ 26 } \cdot (-1) =
7 \cdot -1 = -7 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-5i} + \color{blue}{3i} \color{red}{-1} \color{red}{-7} = \color{blue}{-2i} \color{red}{-8} $$ |
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