Answer
$$i^{23}+i^{18}=-i-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}i^{23}+i^{18}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-i-1\end{aligned} $$ | |
| ① | $$ i^{23} = i^{4 \cdot 5 + 3} =
\left( i^4 \right)^{ 5 } \cdot i^3 =
1^{ 5 } \cdot (-i) =
-i = -i $$ |
| ② | $$ i^{18} = i^{4 \cdot 4 + 2} =
\left( i^4 \right)^{ 4 } \cdot i^2 =
1^{ 4 } \cdot (-1) =
-1 = -1 $$ |
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