Question

which of the following is equivalent to the complex number (i^{17})?
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: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats every 4 powers. So we can use the property of exponents \(a^{m+n}=a^m\times a^n\) to rewrite \(i^{17}\) in terms of a multiple of 4 and a remainder.

Step2: Divide the exponent by 4

We divide 17 by 4: \(17\div4 = 4\) with a remainder of 1. So we can express \(i^{17}\) as \(i^{4\times4 + 1}\).

Step3: Use the property of exponents

Using the property \(a^{mn}=(a^m)^n\) and \(a^{m + n}=a^m\times a^n\), we have \(i^{4\times4+1}=(i^4)^4\times i^1\).

Step4: Substitute the value of \(i^4\)

We know that \(i^4 = 1\), so \((i^4)^4\times i^1=1^4\times i\).

Step5: Simplify the expression

Since \(1^4 = 1\), then \(1^4\times i = i\).

Answer:

B. \(i\)