Question
Answer
$$-8i^{67}-10i^{38}+i^{59}+8i^{15}=-i+10$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-8i^{67}-10i^{38}+i^{59}+8i^{15}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8i+10-i-8i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-i+10\end{aligned} $$ | |
| ① | $$ -8i^{67} = -8 \cdot i^{4 \cdot 16 + 3} =
-8 \cdot \left( i^4 \right)^{ 16 } \cdot i^3 =
-8 \cdot 1^{ 16 } \cdot (-i) =
-8 \cdot -i = 8i $$ |
| ② | $$ -10i^{38} = -10 \cdot i^{4 \cdot 9 + 2} =
-10 \cdot \left( i^4 \right)^{ 9 } \cdot i^2 =
-10 \cdot 1^{ 9 } \cdot (-1) =
-10 \cdot -1 = 10 $$ |
| ③ | $$ i^{59} = i^{4 \cdot 14 + 3} =
\left( i^4 \right)^{ 14 } \cdot i^3 =
1^{ 14 } \cdot (-i) =
-i = -i $$ |
| ④ | $$ 8i^{15} = 8 \cdot i^{4 \cdot 3 + 3} =
8 \cdot \left( i^4 \right)^{ 3 } \cdot i^3 =
8 \cdot 1^{ 3 } \cdot (-i) =
8 \cdot -i = -8i $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{8i}} \, \color{green}{-i} \, \color{green}{ -\cancel{8i}} \,+10 = \color{green}{-i} +10 $$ |
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