Question
Answer
$$4i^{104}-5i^{33}-5i^{102}-12i^{26}=-5i+21$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}4i^{104}-5i^{33}-5i^{102}-12i^{26}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4-5i+5+12 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-5i+21\end{aligned} $$ | |
| ① | $$ 4i^{104} = 4 \cdot i^{4 \cdot 26 + 0} =
4 \cdot \left( i^4 \right)^{ 26 } \cdot i^0 =
4 \cdot 1^{ 26 } \cdot 1 =
4 \cdot 1 $$ |
| ② | $$ -5i^{33} = -5 \cdot i^{4 \cdot 8 + 1} =
-5 \cdot \left( i^4 \right)^{ 8 } \cdot i^1 =
-5 \cdot 1^{ 8 } \cdot i =
-5 \cdot i $$ |
| ③ | $$ -5i^{102} = -5 \cdot i^{4 \cdot 25 + 2} =
-5 \cdot \left( i^4 \right)^{ 25 } \cdot i^2 =
-5 \cdot 1^{ 25 } \cdot (-1) =
-5 \cdot -1 = 5 $$ |
| ④ | $$ -12i^{26} = -12 \cdot i^{4 \cdot 6 + 2} =
-12 \cdot \left( i^4 \right)^{ 6 } \cdot i^2 =
-12 \cdot 1^{ 6 } \cdot (-1) =
-12 \cdot -1 = 12 $$ |
| ⑤ | Combine like terms: $$ -5i+ \color{blue}{4} + \color{red}{5} + \color{red}{12} = -5i+ \color{red}{21} $$ |
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