By fawadtwopointo@gmail.com / July 21, 2025
Ex 1.3 class 10 federal board focuses on Complex Numbers, and Exercise 1.3
continues the journey by diving deeper into the powers of the imaginary unit (i) and simplifying complex expressions.
After building a foundation in Exercise 1.2 with arithmetic operations on complex numbers, this exercise challenges students to apply the identity i² = –1 repeatedly to simplify higher powers of i, and then use those results to simplify algebraic expressions that include complex terms.
Focus of Exercise 1.3
Ex 1.3 class 10 federal board is centered around two major concepts:
- Understanding powers of i
- Simplifying expressions involving complex numbers using those powers
The imaginary unit i has a repeating pattern when raised to powers:
i¹ = i
i² = –1
i³ = –i
i⁴ = 1
i⁵ = i (and the cycle repeats every 4 powers)
Key Topics Covered
Combining like terms in complex expressions after simplification
Identifying the pattern in powers of i
Simplifying higher powers of i using modulo 4 method
Substituting simplified values of iⁿ into complex expressions
Learning Objectives
By the end of Exercise 1.3, students should be able to:
Present complex expressions in the standard form: a + bi
Recognize the cyclic nature of powers of i (every 4 steps)
Efficiently simplify any power of i using the shortcut method (n mod 4)
Substitute and simplify expressions involving higher powers of i
Importance in Board Exams
Exercise 1.3 holds significant value in board exams, particularly in objective (MCQ) and short-answer questions. Students may be asked to:
- Identify the correct simplified value of iⁿ
- Simplify expressions like (2i⁵ + 3i⁷ – i¹⁰)
These types of questions test both conceptual understanding and speed, making this exercise crucial for scoring full marks.
Recommended Practice Strategy
Review past paper MCQs and short questions to strengthen your preparation
Memorize the first four powers of i and understand the repeating pattern
Use the mod 4 rule to simplify iⁿ efficiently (e.g., i²⁷ ≡ i³ = –i)
Solve each question step-by-step, simplifying powers first, then combining terms
Pro Tip for Students
When facing iⁿ, divide n by 4 and look at the remainder:
- Remainder 0 → i⁴ = 1
- Remainder 1 → i
- Remainder 2 → –1
- Remainder 3 → –i
This quick trick can save you time and errors during the exam.
Final Thoughts
Exercise 1.3 helps students build speed and confidence in dealing with imaginary numbers. Once you master the cycle of powers and expression simplification, you’ll find complex number problems much easier to handle in both exams and future algebra topics.
If you want to learn exercise 1.2 then visit our previous blog.
Practice consistently, stay curious, and never skip the logic behind each pattern!
