Question

Evaluate the powers of i:
a) i⁴²
b) i¹⁰⁵

Answer 1

To evaluate the powers of i, we can use the pattern iⁿ = iⁿ⁻⁴, where n is any integer. For i⁴², we can simplify it to i¹⁶, which is equal to 1. For i¹⁰⁵, we can simplify it to (i⁴)²⁵ * i⁵, which is equal to -i.

Step-by-step explanation:

To evaluate the powers of i, we need to first understand that i2 = -1. From this, we can establish a pattern:

1. i1 = i
2. i2 = -1
3. i3 = -i
4. i4 = 1

Using this pattern, we can evaluate the given powers of i:

a) i4² = i16 = 1

b) i105 = i100 * i5 = (i4)25 * i5 = 125 * i5 = i5 = -i