Question 2 (i): Simplify i35 + 1 i31 and Express it in the Form a + ib
Complete Step-by-Step Solution – Exercise 2.1 | Advance Mathematics Class 10 | SEBA
Key Takeaways
- Learn how to simplify higher powers of the imaginary unit i.
- Understand the cyclic pattern of powers of i.
- Simplify reciprocal powers such as 1/in.
- Express the final answer in the standard form a + ib.
- Useful for SEBA, HSLC, CBSE, State Boards, and competitive examinations.
Question
i35 + 1 i31
Express the answer in the form a + ib.
 |
| Question 2. Simplify i^{35}+\frac{1}{i^{31}} - Exercise 2.1 - Advance Math Class10 |
Introduction
The imaginary unit i is defined by the relation i2 = -1. Higher powers of i repeat in a cycle of four, making it easy to simplify large exponents by using these repeating patterns.
In this problem, we simplify the expression i35 + 1 i31 using the basic properties of imaginary numbers and then express the result in the standard form a + ib.
Given Expression
i35 + 1 i31
Basic Properties of i
Property 1
i2 = -1
Property 2
i3 = -i
Property 3
i4 = 1
Repeating Cycle
The powers of i repeat every four powers:
i → -1 → -i → 1
Key Idea
Whenever a large power of i appears, rewrite it using powers of i2 or reduce the exponent modulo 4. This makes simplification quick and accurate.
Step 1: Simplify i35
We know that
i2 = -1
Rewrite i35 as:
i35
= (i2)17 × i
= (-1)17 × i
= -1 × i
= -i
Expression
i35
Rewrite
(i2)17 × i
Use
i2 = -1
Result
-i
Step 2: Simplify
1 i31
Rewrite the denominator as:
i31
= (i2)15 × i
= (-1)15 × i
= -i
Therefore,
1 i31
= 1 -i
Now rationalize the denominator.
= 1 × i -i × i
= i -(i2)
= i -(-1)
= i
Expression
1/i31
Rewrite
1/(-i)
Rationalize
Multiply by i/i
Result
i
Step 3: Combine Both Parts
From the previous steps, we obtained:
i35 = -i
1 i31 = i
Now substitute these values into the given expression.
i35 + 1 i31
= -i + i
= 0
Express the Answer in the Form a + ib
The standard form of a complex number is
a + ib
Since
0 = 0 + 0i
Therefore,
a = 0
b = 0
Calculation Summary
| Expression | Simplified Value |
|---|---|
| i35 | -i |
| 1 i31 | i |
| -i + i | 0 |
| Standard Form | 0 + 0i |
Final Answer
0
or, in the form a + ib, 0 + 0i.
Why Does This Method Work?
The imaginary unit i follows a repeating cycle of four powers. Once this cycle is understood, simplifying higher powers of i becomes very easy.
Fundamental Property
i2 = -1
Using this property,
| Power | Value |
|---|---|
| i1 | i |
| i2 | -1 |
| i3 | -i |
| i4 | 1 |
| i5 | i |
The values repeat after every four powers:
i → -1 → -i → 1 → i → ...
Shortcut Method
Instead of expanding very large powers, divide the exponent by 4 and use the remainder.
Example 1
35 ÷ 4 = 8 remainder 3
i35 = i3 = -i
Example 2
31 ÷ 4 = 7 remainder 3
i31 = i3 = -i
Therefore,
1 i31 = i
Common Mistakes Students Make
❌ Mistake 1
Forgetting that the powers of i repeat after every four terms.
❌ Mistake 2
Using incorrect values such as i2 = 1 instead of -1.
❌ Mistake 3
Not simplifying the reciprocal correctly while evaluating 1/in.
❌ Mistake 4
Forgetting to express the final answer in the required form a + ib.
Memory Tricks
- ✔ Memorize the sequence: i, -1, -i, 1.
- ✔ Divide the exponent by 4 and use the remainder.
- ✔ Remainder 0 → 1
- ✔ Remainder 1 → i
- ✔ Remainder 2 → -1
- ✔ Remainder 3 → -i
Exam Tips
- Always begin with the identity i2 = -1.
- Reduce large powers using the cycle of four.
- Simplify reciprocal terms carefully.
- Write the final answer in the form a + ib if required.
- Recheck the sign before writing the final answer.
Frequently Asked Questions (FAQs)
1. What is the value of i2?
The imaginary unit satisfies the fundamental identity:
i2 = -1
2. Why do the powers of i repeat?
The powers of i repeat after every four exponents because i4 = 1. Therefore, the sequence continues as:
i, -1, -i, 1, i, -1, -i, 1 ...
3. What is the easiest way to simplify large powers of i?
Divide the exponent by 4 and use the remainder.
- Remainder 0 → 1
- Remainder 1 → i
- Remainder 2 → -1
- Remainder 3 → -i
4. What is the value of i35?
Since 35 leaves a remainder of 3 when divided by 4,
i35 = i3 = -i
5. What is the value of 1/i31?
Since i31 = -i,
1/i31 = 1/(-i) = i
6. What is the final answer?
i35 + 1 i31
= -i + i
= 0
7. What is the answer in the form a + ib?
The standard form is
0 + 0i
Quick Revision
- ✔ i2 = -1
- ✔ Powers of i repeat every four terms.
- ✔ i35 = -i
- ✔ 1/i31 = i
- ✔ -i + i = 0
- ✔ Standard form = 0 + 0i
Summary Table
| Expression | Value |
|---|---|
| i2 | -1 |
| i35 | -i |
| 1 i31 | i |
| -i + i | 0 |
| Answer in a + ib Form | 0 + 0i |
Conclusion
By applying the cyclic properties of the imaginary unit i, we simplified both terms of the given expression separately. After evaluating i35 and 1/i31, the two terms cancelled each other, giving a final result of zero.
Therefore, the simplified expression is 0, which can also be written in the standard complex-number form as 0 + 0i.
Final Answer
i35 + 1 i31
= 0
In the form a + ib: 0 + 0i
References
- SEBA Class 10 Advance Mathematics Textbook
- NCERT Mathematics – Complex Numbers
- CBSE Mathematics Curriculum
- Standard Algebra Textbooks
- Properties of Imaginary Numbers and Complex Numbers