Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Problems climb from reading a point to roots of unity, alternating current and robot moves. Use exact values unless a decimal is requested.
Standards: N-CN.B.4, N-CN.B.5, N-CN.B.6, N-CN.A.3 · Level: Grade 12 Precalculus
Problem 1 Modulus of a Complex Number ★★★
Find the modulus of z = 2 − 5i. Give an exact value and a decimal rounded to hundredths.
Problem 2 Conjugate and Product ★★★
Let z = 3 − 7i. Write the conjugate z̄, then compute z·z̄ and compare it with |z|2.
Problem 3 Reading a Point ★★★
Write the complex number z shown in the figure, state its conjugate, and find its modulus.
Problem 4 Modulus and Argument ★★★
Find the modulus and the argument (in degrees, 0° ≤ θ < 360°) of −5 + 5i.
Problem 5 Distance and Midpoint ★★★
Points P and Q in the figure represent −3 + 5i and 4 + i. Find the distance PQ and the midpoint of the segment PQ.
Problem 6 Polar to Rectangular ★★★
Write 6(cos 60° + i sin 60°) in the form a + bi.
Problem 7 Rectangular to Polar ★★★
The figure shows z = 2 − 2√3i. Write z in polar form with 0° ≤ θ < 360°.
Problem 8 Multiplying in Polar Form ★★★
Multiply 4(cos 25° + i sin 25°) by 5(cos 35° + i sin 35°). Give the result in polar form and in the form a + bi.
Problem 9 Dividing in Polar Form ★★★
Divide 18(cos 200° + i sin 200°) by 6(cos 80° + i sin 80°) and write the quotient in the form a + bi.
Problem 10 Product via Polar Form ★★★
Let z₁ = 1 + i and z₂ = 1 + √3i. Write each in polar form, then give the modulus and an argument of z₁z₂.
Problem 11 A Tenth Power ★★★
Write 1 − i in polar form first, then evaluate (1 − i)10 with De Moivre’s theorem.
Problem 12 Cubing in Polar Form ★★★
Evaluate [2(cos 20° + i sin 20°)]3 and write the result in the form a + bi.
Problem 13 A Sixth Power ★★★
Evaluate (√3 + i)6.
Problem 14 A Fifth Power ★★★
Evaluate (−1 + i)5 and write the answer in the form a + bi.
Problem 15 Cube Roots of 64 ★★★
Find all three cube roots of 64 and write them in the form a + bi.
Problem 16 Trail Distance ★★★
On a park map, each unit of the complex plane is 1 mile. Trailhead A is at −2 + 6i and trailhead B is at 5 − 3i. A rest stop is built at the midpoint. Find the straight-line distance from A to B to the nearest hundredth of a mile and the location of the rest stop.
Problem 17 Robot Rotation ★★★
A warehouse robot starts at 3 + 2i (in meters). Each move multiplies its position by 2i, which doubles its distance from the origin and turns it 90° counterclockwise. (a) Find its position after one move. (b) Find its position after four identical moves.
Problem 18 Alternating Current ★★★
In an AC circuit, a voltage of 48(cos 70° + i sin 70°) volts is applied to an impedance of 6(cos 25° + i sin 25°) ohms. The current is voltage ÷ impedance. Find the current in polar form and in the form a + bi (decimals to hundredths).
Problem 19 Sixth Roots of Unity ★★★
(a) Show that z = cos 60° + i sin 60° satisfies z6 = 1. (b) List all six sixth roots of 1 in the form a + bi. (c) Describe the polygon they form.
Problem 20 A Quotient of Powers ★★★
Simplify (1 − √3i)5 ÷ (1 + i)4 and write the result in the form a + bi.
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