Question
Answer
$$-5i^{18}-6i^{33}+3i^{46}+2i^{53}=-4i+2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-5i^{18}-6i^{33}+3i^{46}+2i^{53}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}5-6i-3+2i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4i+2\end{aligned} $$ | |
| ① | $$ -5i^{18} = -5 \cdot i^{4 \cdot 4 + 2} =
-5 \cdot \left( i^4 \right)^{ 4 } \cdot i^2 =
-5 \cdot 1^{ 4 } \cdot (-1) =
-5 \cdot -1 = 5 $$ |
| ② | $$ -6i^{33} = -6 \cdot i^{4 \cdot 8 + 1} =
-6 \cdot \left( i^4 \right)^{ 8 } \cdot i^1 =
-6 \cdot 1^{ 8 } \cdot i =
-6 \cdot i $$ |
| ③ | $$ 3i^{46} = 3 \cdot i^{4 \cdot 11 + 2} =
3 \cdot \left( i^4 \right)^{ 11 } \cdot i^2 =
3 \cdot 1^{ 11 } \cdot (-1) =
3 \cdot -1 = -3 $$ |
| ④ | $$ 2i^{53} = 2 \cdot i^{4 \cdot 13 + 1} =
2 \cdot \left( i^4 \right)^{ 13 } \cdot i^1 =
2 \cdot 1^{ 13 } \cdot i =
2 \cdot i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-6i} + \color{blue}{2i} \color{red}{-3} + \color{red}{5} = \color{blue}{-4i} + \color{red}{2} $$ |
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