Perform the indicated operations and simplify:

i^-31

i^-31

= 1/i^31
= 1/( i^28 * i^3)
= 1/((1)(-i)
= -1/i
or
= -i^-1

To simplify the expression i^-31, we first need to understand the properties of the imaginary unit i. The imaginary unit i is defined as the square root of -1. Therefore, i * i = -1.

Now, let's simplify the expression i^-31 step by step:

1. Start with i^-1. We know that any number raised to the power of -1 is equal to its reciprocal. So,
i^-1 = 1/i

2. Next, simplify the expression i^-1 = 1/i by multiplying the numerator and denominator by i:
1/i = (i * 1) / (i * i) = i / (i * i)

3. Apply the property i * i = -1:
i / (i * i) = i / (-1) = -i

Therefore, the simplified expression of i^-31 is -i.