Powers of i: Class 11 practice
Powers of i repeat every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Division with remainders
The idea behind powers of i
Powers of i repeat every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Divide the exponent by 4 and use the remainder. Negative powers: i⁻¹ = −i.
Step-by-step method
- Divide the exponent by 4.
- Use the remainder to read off the value.
- Add or simplify.
Worked example 1 (Basic)
Find i^158.
- 158 = 4 × 39 + 2
- i^158 = i^2 = −1
- Answer: −1
Worked example 2 (Standard)
Simplify i^136 + i^152.
- 136 mod 4 = 0 → 1
- 152 mod 4 = 0 → 1
- = 2
- Answer: 2
Worked example 3 (Challenge)
Find i^8 + i^9 + i^10 + i^11.
- 8 is a multiple of 4, so the terms are 1, i, −1, −i
- = 0
- Answer: 0
Common mistakes and misconceptions
- Using the quotient instead of the remainder
- Forgetting i⁰ = 1 for remainder 0
Practice questions
- (Basic) Find i^160.
- (Basic) Find i^93.
- (Standard) Simplify i^103 + i^182.
- (Standard) Simplify i^26 + i^65.
- (Challenge) Find i^152 + i^153 + i^154 + i^155.
- (Challenge) Find i^48 + i^49 + i^50 + i^51.
Answer key with full working
- Q1: 160 = 4 × 40 + 0 → i^160 = i^0 = 1 → Answer: 1
- Q2: 93 = 4 × 23 + 1 → i^93 = i^1 = i → Answer: i
- Q3: 103 mod 4 = 3 → −i → 182 mod 4 = 2 → −1 → = -1 − i → Answer: -1 − i
- Q4: 26 mod 4 = 2 → −1 → 65 mod 4 = 1 → i → = -1 + i → Answer: -1 + i
- Q5: 152 is a multiple of 4, so the terms are 1, i, −1, −i → = 0 → Answer: 0
- Q6: 48 is a multiple of 4, so the terms are 1, i, −1, −i → = 0 → Answer: 0
Syllabus note
This skill is listed under "Complex Numbers" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is powers of i in?
"Complex Numbers", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.