Question
Answer
$$-6i^{62}+10i^{113}-i^{22}-4i^{55}=14i+7$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-6i^{62}+10i^{113}-i^{22}-4i^{55}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6+10i+1+4i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}14i+7\end{aligned} $$ | |
| ① | $$ -6i^{62} = -6 \cdot i^{4 \cdot 15 + 2} =
-6 \cdot \left( i^4 \right)^{ 15 } \cdot i^2 =
-6 \cdot 1^{ 15 } \cdot (-1) =
-6 \cdot -1 = 6 $$ |
| ② | $$ 10i^{113} = 10 \cdot i^{4 \cdot 28 + 1} =
10 \cdot \left( i^4 \right)^{ 28 } \cdot i^1 =
10 \cdot 1^{ 28 } \cdot i =
10 \cdot i $$ |
| ③ | $$ -i^{22} = - i^{4 \cdot 5 + 2} =
- \left( i^4 \right)^{ 5 } \cdot i^2 =
- 1^{ 5 } \cdot (-1) =
- -1 = 1 $$ |
| ④ | $$ -4i^{55} = -4 \cdot i^{4 \cdot 13 + 3} =
-4 \cdot \left( i^4 \right)^{ 13 } \cdot i^3 =
-4 \cdot 1^{ 13 } \cdot (-i) =
-4 \cdot -i = 4i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{10i} + \color{blue}{4i} + \color{red}{1} + \color{red}{6} = \color{blue}{14i} + \color{red}{7} $$ |
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