Question

which of the following is equivalent to ( i^{31} )?
a. (-1)
b. (i)
c. (1)
d. (-i)

Explanation:

Step1: Recall the cycle of \(i\)

The imaginary unit \(i\) has a cyclic property: \(i^1 = i\), \(i^2 = -1\), \(i^3 = i^2 \cdot i = -i\), \(i^4 = (i^2)^2 = (-1)^2 = 1\), and then the cycle repeats every 4 powers. So we can express \(i^{31}\) in terms of a multiple of 4 plus a remainder.

Step2: Divide 31 by 4

We divide 31 by 4: \(31\div4 = 7\) with a remainder of 3. So we can write \(i^{31}=i^{4\times7 + 3}\).

Step3: Use the property of exponents

Using the property \(a^{m + n}=a^m\cdot a^n\), we have \(i^{4\times7 + 3}=(i^4)^7\cdot i^3\). Since \(i^4 = 1\), this simplifies to \(1^7\cdot i^3\). And since \(1^7 = 1\), we have \(i^3\).

Step4: Recall the value of \(i^3\)

We know that \(i^3 = i^2\cdot i=-1\cdot i=-i\).

Answer:

D. \(-i\)