Question

which of the following is equivalent to the complex number $i^{36}$? choose 1 answer: a 1 b $i$ c $-1$ d $-i$

Explanation:

Step1: Recall the cycle of \(i\)

The imaginary unit \(i\) has a cyclic pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = i^2 \cdot i = -i\), \(i^4 = (i^2)^2 = (-1)^2 = 1\), and then the pattern repeats every 4 powers.

Step2: Divide the exponent by 4

To find \(i^{36}\), we divide the exponent 36 by 4: \(36 \div 4 = 9\) with a remainder of 0.

Step3: Use the cycle property

Since the remainder is 0, \(i^{36}\) is equivalent to \(i^{4 \times 9} = (i^4)^9\). We know \(i^4 = 1\), so \((i^4)^9 = 1^9 = 1\).

Answer:

A. 1