Complex Numbers · Sub-skill drill
Powers of i
The imaginary unit i is defined by i² = −1. The powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then the cycle repeats. To compute i^n for any positive integer n, divide n by 4 and look up the remainder in the cycle: remainder 1 gives i, remainder 2 gives −1, remainder 3 gives −i, remainder 0 gives 1. SAT questions on powers of i are quick once you recognize the four-step cycle.
How this sub-skill is tested on the SAT
The imaginary unit i is defined by i² = −1. The powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then the cycle repeats. To compute i^n for any positive integer n, divide n by 4 and look up the remainder in the cycle: remainder 1 gives i, remainder 2 gives −1, remainder 3 gives −i, remainder 0 gives 1. SAT questions on powers of i are quick once you recognize the four-step cycle.
This sub-skill sits inside the broader Complex Numbers topic, which is part of the College Board's Additional Topics in Math content domain. Additional Topics in Math is the smallest official SAT Math domain by raw question count, but it carries outsized weight because the questions are concentrated at the harder end of each section. You will see roughly six of these per test, and they tend to separate students aiming for a 750 from students aiming for an 800. The domain covers right triangle trigonometry, circle theorems, volume formulas, complex number arithmetic, and the geometry of lines in the coordinate plane. Most of the formulas you need are listed at the start of the math section — but the test rewards students who have
Practice questions in this drill set
Below are 7 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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What is the product (-6 + 5i)(3 + 5i) expressed in the form a + bi?
- A -43 - 15i
- B -18 + 25i
- C -42 - 15i
- D -43 + -14i
Worked solution
Answer: A — -43 - 15i
Use FOIL and replace i^2 with -1. (-6 + 5i)(3 + 5i) = -18 + -30i + 15i + 25i^2 = -18 + -15i + 25(-1) = -43 + -15i.
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What is the product (-6 + 4i)(-1 - 1i) expressed in the form a + bi?
- A 11 + 2i
- B 10 + 3i
- C 10 + 2i
- D 6 + -4i
Worked solution
Answer: C — 10 + 2i
Use FOIL and replace i^2 with -1. (-6 + 4i)(-1 + -1i) = 6 + 6i + -4i + -4i^2 = 6 + 2i + -4(-1) = 10 + 2i.
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What is the product (4 + 6i)(-2 - 2i) expressed in the form a + bi?
- A 4 - 20i
- B 5 - 20i
- C 4 + -19i
- D -8 + -12i
Worked solution
Answer: A — 4 - 20i
Use FOIL and replace i^2 with -1. (4 + 6i)(-2 + -2i) = -8 + -8i + -12i + -12i^2 = -8 + -20i + -12(-1) = 4 + -20i.
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What is the product (-6 - 2i)(-1 + 6i) expressed in the form a + bi?
- A 6 + -12i
- B 19 - 34i
- C 18 + -33i
- D 18 - 34i
Worked solution
Answer: D — 18 - 34i
Use FOIL and replace i^2 with -1. (-6 + -2i)(-1 + 6i) = 6 + -36i + 2i + -12i^2 = 6 + -34i + -12(-1) = 18 + -34i.
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What is the product (2 - 1i)(1 + 5i) expressed in the form a + bi?
- A 8 + 9i
- B 2 + -5i
- C 7 + 10i
- D 7 + 9i
Worked solution
Answer: D — 7 + 9i
Use FOIL and replace i^2 with -1. (2 + -1i)(1 + 5i) = 2 + 10i + -1i + -5i^2 = 2 + 9i + -5(-1) = 7 + 9i.
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What is the product (-1 - 1i)(-3 - 1i) expressed in the form a + bi?
- A 2 + 5i
- B 3 + 4i
- C 2 + 4i
- D 3 + 1i
Worked solution
Answer: C — 2 + 4i
Use FOIL and replace i^2 with -1. (-1 + -1i)(-3 + -1i) = 3 + 1i + 3i + 1i^2 = 3 + 4i + 1(-1) = 2 + 4i.
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What is the product (1 - 5i)(1 - 1i) expressed in the form a + bi?
- A -3 - 6i
- B -4 - 6i
- C 1 + 5i
- D -4 + -5i
Worked solution
Answer: B — -4 - 6i
Use FOIL and replace i^2 with -1. (1 + -5i)(1 + -1i) = 1 + -1i + -5i + 5i^2 = 1 + -6i + 5(-1) = -4 + -6i.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.