Question
Answer
$$3i^5+9i^{21}-12i^{46}+8i^{105}=20i+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3i^5+9i^{21}-12i^{46}+8i^{105}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3i+9i+12+8i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}20i+12\end{aligned} $$ | |
| ① | $$ 3i^5 = 3 \cdot i^{4 \cdot 1 + 1} =
3 \cdot \left( i^4 \right)^{ 1 } \cdot i^1 =
3 \cdot 1^{ 1 } \cdot i =
3 \cdot i $$ |
| ② | $$ 9i^{21} = 9 \cdot i^{4 \cdot 5 + 1} =
9 \cdot \left( i^4 \right)^{ 5 } \cdot i^1 =
9 \cdot 1^{ 5 } \cdot i =
9 \cdot i $$ |
| ③ | $$ -12i^{46} = -12 \cdot i^{4 \cdot 11 + 2} =
-12 \cdot \left( i^4 \right)^{ 11 } \cdot i^2 =
-12 \cdot 1^{ 11 } \cdot (-1) =
-12 \cdot -1 = 12 $$ |
| ④ | $$ 8i^{105} = 8 \cdot i^{4 \cdot 26 + 1} =
8 \cdot \left( i^4 \right)^{ 26 } \cdot i^1 =
8 \cdot 1^{ 26 } \cdot i =
8 \cdot i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{3i} + \color{red}{9i} + \color{red}{8i} +12 = \color{red}{20i} +12 $$ |
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