Question
Answer
$$12i^{116}-10i^{90}-3i^{111}+8i^{50}=3i+14$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}12i^{116}-10i^{90}-3i^{111}+8i^{50}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}12+10+3i-8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3i+14\end{aligned} $$ | |
| ① | $$ 12i^{116} = 12 \cdot i^{4 \cdot 29 + 0} =
12 \cdot \left( i^4 \right)^{ 29 } \cdot i^0 =
12 \cdot 1^{ 29 } \cdot 1 =
12 \cdot 1 $$ |
| ② | $$ -10i^{90} = -10 \cdot i^{4 \cdot 22 + 2} =
-10 \cdot \left( i^4 \right)^{ 22 } \cdot i^2 =
-10 \cdot 1^{ 22 } \cdot (-1) =
-10 \cdot -1 = 10 $$ |
| ③ | $$ -3i^{111} = -3 \cdot i^{4 \cdot 27 + 3} =
-3 \cdot \left( i^4 \right)^{ 27 } \cdot i^3 =
-3 \cdot 1^{ 27 } \cdot (-i) =
-3 \cdot -i = 3i $$ |
| ④ | $$ 8i^{50} = 8 \cdot i^{4 \cdot 12 + 2} =
8 \cdot \left( i^4 \right)^{ 12 } \cdot i^2 =
8 \cdot 1^{ 12 } \cdot (-1) =
8 \cdot -1 = -8 $$ |
| ⑤ | Combine like terms: $$ 3i+ \color{blue}{10} + \color{red}{12} \color{red}{-8} = 3i+ \color{red}{14} $$ |
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