Question
Answer
$$-4i^{61}-2i^{41}-4i^{58}-4i^{107}=-2i+4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-4i^{61}-2i^{41}-4i^{58}-4i^{107}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-4i-2i+4+4i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-2i+4\end{aligned} $$ | |
| ① | $$ -4i^{61} = -4 \cdot i^{4 \cdot 15 + 1} =
-4 \cdot \left( i^4 \right)^{ 15 } \cdot i^1 =
-4 \cdot 1^{ 15 } \cdot i =
-4 \cdot i $$ |
| ② | $$ -2i^{41} = -2 \cdot i^{4 \cdot 10 + 1} =
-2 \cdot \left( i^4 \right)^{ 10 } \cdot i^1 =
-2 \cdot 1^{ 10 } \cdot i =
-2 \cdot i $$ |
| ③ | $$ -4i^{58} = -4 \cdot i^{4 \cdot 14 + 2} =
-4 \cdot \left( i^4 \right)^{ 14 } \cdot i^2 =
-4 \cdot 1^{ 14 } \cdot (-1) =
-4 \cdot -1 = 4 $$ |
| ④ | $$ -4i^{107} = -4 \cdot i^{4 \cdot 26 + 3} =
-4 \cdot \left( i^4 \right)^{ 26 } \cdot i^3 =
-4 \cdot 1^{ 26 } \cdot (-i) =
-4 \cdot -i = 4i $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ -\cancel{4i}} \, \color{green}{-2i} + \, \color{green}{ \cancel{4i}} \,+4 = \color{green}{-2i} +4 $$ |
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