Question

Match each complex number with its equivalent expression.

i
-1
1
-i

i^48
i^223
i^113
i^82

Answer 1

Remember that i = √(-1), so that

• i ² = -1,

• i ³ = i x i ² = i x (-1) = -i

• i ⁴ = i x i ³ = i x (-i) = -i ² = -(-1) = 1

Multiplying by i again gives i ⁵ = i and the cycle starts over.

The main observation to make here is that raising i to any multiple of 4 returns 1. So,

i ⁴⁸ = (i ⁴)¹² = 1¹² = 1

i ²²³ = i ²²⁰⁺³ = i ²²⁰ x i ³ = (i ⁴)⁵⁵ x i ³ = 1⁵⁵ x (-i) = -i

i ¹¹³ = i ¹¹²⁺¹ = i ¹¹² x i = (i ⁴)²⁸ x i = 1²⁸ x i = i

i ⁸² = i ⁸⁰⁺² = i ⁸⁰ x i ² = (i ⁴)²⁰ x i ² = 1²⁰ x (-1) = -1

Answer 2

i^48 = 1

i^223 = -i

i^113 = i

i^82 = -1