Step by Step
1
Know what i is
i is defined as the square root of −1, so i² = −1. This lets you work with square roots of negative numbers, which have no real-number solution otherwise.
2
Memorize the 4-cycle of powers
Powers of i repeat in a cycle of four: i¹=i, i²=−1, i³=−i, i⁴=1, then it repeats. For any higher power, divide the exponent by 4 and use the remainder to find the equivalent smaller power.
3
Multiply complex numbers using FOIL
Complex numbers in the form (a + bi) multiply just like binomials — use FOIL, then replace any i² that shows up with −1.
4
Use the conjugate to divide
To divide by a complex number, multiply both the numerator and denominator by the conjugate of the denominator — (a − bi) is the conjugate of (a + bi); only the sign on the imaginary part flips, the real part stays the same.
Applied Walkthrough
1
Find i¹³. Divide the exponent by 4: 13 ÷ 4 = 3 remainder 1. So i¹³ behaves like i¹ = i.
2
Multiply (3 + 2i)(4 − i) using FOIL: 12 − 3i + 8i − 2i².
3
Replace i² with −1: −2i² becomes −2(−1) = +2. Combine: 12 + 5i + 2 = 14 + 5i.
4
If you needed to divide by (4 − i) instead, you'd multiply top and bottom by its conjugate, (4 + i), to eliminate the imaginary part from the denominator.
Exam Application
Exams test whether you can correctly reduce higher powers of i using the 4-cycle, whether you remember to replace every i² with −1 after multiplying complex numbers, and whether you can correctly identify a conjugate.
⚠ Common Trap
The most common trap is leaving i² in a final answer instead of replacing it with −1 — every single i² must be converted before the answer is complete. A second common trap is getting the conjugate wrong; only the imaginary part's sign flips, the real part never changes.
✓ Quick Self-Check
1. What does i² equal?
−1.
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2. Find i¹³.
i (since 13 ÷ 4 leaves remainder 1, and i¹ = i).
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3. What is the conjugate of (3 − 2i)?
(3 + 2i) — only the imaginary part's sign flips.
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4. Multiply (3 + 2i)(4 − i).
14 + 5i.
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5. What must you always do with i² after multiplying complex numbers?
Replace it with −1 before finalizing the answer.
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