Answer
$$3i^{44}+11i^{28}-4i^{67}+11i^{116}=4i+25$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3i^{44}+11i^{28}-4i^{67}+11i^{116}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3+11+4i+11 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4i+25\end{aligned} $$ | |
| ① | $$ 3i^{44} = 3 \cdot i^{4 \cdot 11 + 0} =
3 \cdot \left( i^4 \right)^{ 11 } \cdot i^0 =
3 \cdot 1^{ 11 } \cdot 1 =
3 \cdot 1 $$ |
| ② | $$ 11i^{28} = 11 \cdot i^{4 \cdot 7 + 0} =
11 \cdot \left( i^4 \right)^{ 7 } \cdot i^0 =
11 \cdot 1^{ 7 } \cdot 1 =
11 \cdot 1 $$ |
| ③ | $$ -4i^{67} = -4 \cdot i^{4 \cdot 16 + 3} =
-4 \cdot \left( i^4 \right)^{ 16 } \cdot i^3 =
-4 \cdot 1^{ 16 } \cdot (-i) =
-4 \cdot -i = 4i $$ |
| ④ | $$ 11i^{116} = 11 \cdot i^{4 \cdot 29 + 0} =
11 \cdot \left( i^4 \right)^{ 29 } \cdot i^0 =
11 \cdot 1^{ 29 } \cdot 1 =
11 \cdot 1 $$ |
| ⑤ | Combine like terms: $$ 4i+ \color{blue}{11} + \color{red}{3} + \color{red}{11} = 4i+ \color{red}{25} $$ |
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