Question

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

Explanation:

Step1: Find remainder of exponent mod 4

$31 \div 4 = 7$ remainder $3$, so $i^{31}=i^{4\times7 + 3}=(i^4)^7 \times i^3$

Step2: Use $i^4=1$ to simplify

$(1)^7 \times i^3 = 1 \times i^3$

Step3: Simplify $i^3$ using $i^2=-1$

$i^3 = i^2 \times i = -1 \times i = -i$

Answer:

D. -i