Question
Answer
$$-5i^{42}+2i^{119}+12i^{100}-2i^{53}=-4i+17$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-5i^{42}+2i^{119}+12i^{100}-2i^{53}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}5-2i+12-2i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4i+17\end{aligned} $$ | |
| ① | $$ -5i^{42} = -5 \cdot i^{4 \cdot 10 + 2} =
-5 \cdot \left( i^4 \right)^{ 10 } \cdot i^2 =
-5 \cdot 1^{ 10 } \cdot (-1) =
-5 \cdot -1 = 5 $$ |
| ② | $$ 2i^{119} = 2 \cdot i^{4 \cdot 29 + 3} =
2 \cdot \left( i^4 \right)^{ 29 } \cdot i^3 =
2 \cdot 1^{ 29 } \cdot (-i) =
2 \cdot -i = -2i $$ |
| ③ | $$ 12i^{100} = 12 \cdot i^{4 \cdot 25 + 0} =
12 \cdot \left( i^4 \right)^{ 25 } \cdot i^0 =
12 \cdot 1^{ 25 } \cdot 1 =
12 \cdot 1 $$ |
| 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: $$ \color{blue}{-2i} \color{blue}{-2i} + \color{red}{12} + \color{red}{5} = \color{blue}{-4i} + \color{red}{17} $$ |
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