Answer
$$-i^{95}-2i^{35}-12i^{86}+11i^{47}=-8i+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-i^{95}-2i^{35}-12i^{86}+11i^{47}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}i+2i+12-11i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-8i+12\end{aligned} $$ | |
| ① | $$ -i^{95} = - i^{4 \cdot 23 + 3} =
- \left( i^4 \right)^{ 23 } \cdot i^3 =
- 1^{ 23 } \cdot (-i) =
- -i = i $$ |
| ② | $$ -2i^{35} = -2 \cdot i^{4 \cdot 8 + 3} =
-2 \cdot \left( i^4 \right)^{ 8 } \cdot i^3 =
-2 \cdot 1^{ 8 } \cdot (-i) =
-2 \cdot -i = 2i $$ |
| ③ | $$ -12i^{86} = -12 \cdot i^{4 \cdot 21 + 2} =
-12 \cdot \left( i^4 \right)^{ 21 } \cdot i^2 =
-12 \cdot 1^{ 21 } \cdot (-1) =
-12 \cdot -1 = 12 $$ |
| ④ | $$ 11i^{47} = 11 \cdot i^{4 \cdot 11 + 3} =
11 \cdot \left( i^4 \right)^{ 11 } \cdot i^3 =
11 \cdot 1^{ 11 } \cdot (-i) =
11 \cdot -i = -11i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{i} + \color{red}{2i} \color{red}{-11i} +12 = \color{red}{-8i} +12 $$ |
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