Question
Answer
$$-6i^{58}+8i^{108}+2i^{75}+10i^{89}=8i+14$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-6i^{58}+8i^{108}+2i^{75}+10i^{89}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6+8-2i+10i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}8i+14\end{aligned} $$ | |
| ① | $$ -6i^{58} = -6 \cdot i^{4 \cdot 14 + 2} =
-6 \cdot \left( i^4 \right)^{ 14 } \cdot i^2 =
-6 \cdot 1^{ 14 } \cdot (-1) =
-6 \cdot -1 = 6 $$ |
| ② | $$ 8i^{108} = 8 \cdot i^{4 \cdot 27 + 0} =
8 \cdot \left( i^4 \right)^{ 27 } \cdot i^0 =
8 \cdot 1^{ 27 } \cdot 1 =
8 \cdot 1 $$ |
| ③ | $$ 2i^{75} = 2 \cdot i^{4 \cdot 18 + 3} =
2 \cdot \left( i^4 \right)^{ 18 } \cdot i^3 =
2 \cdot 1^{ 18 } \cdot (-i) =
2 \cdot -i = -2i $$ |
| ④ | $$ 10i^{89} = 10 \cdot i^{4 \cdot 22 + 1} =
10 \cdot \left( i^4 \right)^{ 22 } \cdot i^1 =
10 \cdot 1^{ 22 } \cdot i =
10 \cdot i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-2i} + \color{blue}{10i} + \color{red}{6} + \color{red}{8} = \color{blue}{8i} + \color{red}{14} $$ |
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