ACT Math: Evaluate 2i⁴³ - 3i⁸² + i¹⁰⁵ Using Cyclic Powers of i

Exact Question Text

For the imaginary unit $i$ where $i = \sqrt{-1}$, what is the value of the expression $2i^{43} - 3i^{82} + i^{105}$?

Step-by-Step Solution

1. Cycle of $i$ (Divide exponent by 4):

• $43 \div 4 = 10\text{ R }3 \implies i^{43} = i^3 = -i$
• $82 \div 4 = 20\text{ R }2 \implies i^{82} = i^2 = -1$
• $105 \div 4 = 26\text{ R }1 \implies i^{105} = i^1 = i$

1. Substitute & Combine:

$$ 2(-i) - 3(-1) + i = -2i + 3 + i = \mathbf{3 - i} $$