Question
Answer
$$i^5+3^4-5^3+6i^2-3i^5+3i^4-5i^3+6i^2-3i=-53$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}i^5+3^4-5^3+6i^2-3i^5+3i^4-5i^3+6i^2-3i& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}i+3^4-5^3+(-6)-3i+3--5i+(-6)-3i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}i+81-125+(-6)-3i+3--5i+(-6)-3i \xlongequal{ } \\[1 em] & \xlongequal{ }i+81-125-6-3i+3+5i-6-3i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-53\end{aligned} $$ | |
| ① | $$ i^5 = i^{4 \cdot 1 + 1} =
\left( i^4 \right)^{ 1 } \cdot i^1 =
1^{ 1 } \cdot i =
i $$$$ 6i^2 = 6 \cdot (-1) = -6 $$$$ 3i^5 = 3 \cdot i^{4 \cdot 1 + 1} =
3 \cdot \left( i^4 \right)^{ 1 } \cdot i^1 =
3 \cdot 1^{ 1 } \cdot i =
3 \cdot i $$$$ 3i^4 = 3 \cdot i^2 \cdot i^2 =
3 \cdot ( - 1) \cdot ( - 1) =
3 $$$$ 5i^3 = 5 \cdot \color{blue}{i^2} \cdot i =
5 \cdot ( \color{blue}{-1}) \cdot i =
-5 \cdot \, i $$$$ 6i^2 = 6 \cdot (-1) = -6 $$ |
| ② | 3i-3i=0i3i-3i=0i |
| ③ | Combine like terms: $$ \color{blue}{i} + \color{red}{81} \color{green}{-125} \color{orange}{-6} \color{blue}{-3i} + \color{red}{3} + \color{green}{5i} \color{red}{-6} \color{green}{-3i} = \color{red}{-53} $$ |
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