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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Every solution below shows the key steps and the final answer in bold. Compare each method with the work on the worksheet, paying attention to where i2 is replaced by −1 and to sign errors.
Standards: N-RN.A.1, N-RN.A.2, N-CN.A.1, N-CN.A.2 · Level: Grade 11 Algebra 2
Problem 1 Sorting numbers into sets
a) 0.375 = 38: rational, real.
b) √45 = 3√5 is not a perfect square root: irrational, real.
c) −√64 = −8: integer, rational, real (not natural).
d) 59 is a quotient of integers: rational, real.
e) √(−25) = 5i: non-real (complex).
Problem 2 Simplifying negative roots
a) √(−81) = 9i.
b) √(−44) = √4 · √11 · i = 2i√11.
c) √(−72) = 6i√2, so the opposite is −6i√2.
Problem 3 Reading a point on the complex plane
The point is 4 units right and 3 units down, so Z = 4 − 3i.
Real part 4, imaginary part −3. The conjugate is 4 + 3i.
Problem 4 Adding and subtracting
a) Real: 6 + (−9) = −3. Imaginary: 2 + 7 = 9. −3 + 9i
b) 4 − 11i + 5 + 6i = 9 − 5i.
Problem 5 Identifying parts
a) Real 9, imaginary −4.
b) Real 0, imaginary −7.
c) Real 12, imaginary 0.
d) Real 12, imaginary 34.
(a) 9 and −4; (b) 0 and −7; (c) 12 and 0; (d) 12 and 34
Problem 6 Plotting and naming points
Read the horizontal position, then the vertical position.
A = 5 − 2i, B = −3 + 4i, C = −1 − 3i, D = 4i
Problem 7 Multiplying binomials
FOIL: 20 − 30i + 12i − 18i2.
Replace i2 by −1: 20 − 18i + 18 = 38 − 18i.
Problem 8 Products with conjugates
a) −21i2 = 21.
b) 4 + 2i − 2i − i2 = 4 + 1 = 5.
c) 25 − 36i2 = 25 + 36 = 61.
(a) 21; (b) 5; (c) 61
Problem 9 Powers of an imaginary number
a) 9i2 = −9.
b) 32i5 = 32i (since i5 = i).
c) −64i3 = −64(−i) = 64i.
(a) −9; (b) 32i; (c) 64i
Problem 10 Product of negative roots
a) 3i · 4i = 12i2 = −12.
b) i√5 · 2i√5 = 10i2 = −10.
(a) −12; (b) −10
Problem 11 Conjugate product
a) Conjugate 8 + 3i. Product 64 − 9i2 = 64 + 9 = 73.
b) Conjugate −5i. Product −25i2 = 25.
(a) 8 + 3i and 73; (b) −5i and 25
Problem 12 Equal complex numbers
Two complex numbers are equal when their real parts match and their imaginary parts match.
x + 3 = 9 gives x = 6. 2y − 1 = 7 gives y = 4.
x = 6, y = 4
Problem 13 Adding on the complex plane
u + v = (4 − 2) + (1 + 3)i = 2 + 4i.
u + v = 2 + 4i
Problem 14 Squaring a binomial
(1 − i)2 = 1 − 2i + i2 = 1 − 2i − 1 = −2i.
Then (1 − i)6 = (−2i)3 = −8i3 = −8(−i) = 8i.
(1 − i)6 = 8i
Problem 15 Checking a solution
(3 + 5i)2 = 9 + 30i − 25 = −16 + 30i.
−6(3 + 5i) = −18 − 30i.
Sum: −16 + 30i − 18 − 30i + 34 = 0.
Yes, 3 + 5i is a solution.
Problem 16 Sum of four consecutive powers
The remainders on division by 4 are 1, 2, 3, 0, so the values are i, −1, −i, 1.
Sum: i − 1 − i + 1 = 0.
Problem 17 Series impedance
Series impedances add: (12 + 7) + (5 − 9)i.
19 − 4i ohms
Problem 18 Voltage in an AC circuit
(5 + 2i)(3 − 6i) = 15 − 30i + 6i − 12i2.
= 15 − 24i + 12 = 27 − 24i.
27 − 24i volts
Problem 19 Quarter turns of a marker
(3 + 2i)i = 3i + 2i2 = −2 + 3i.
(−2 + 3i)i = −3 − 2i.
(−3 − 2i)i = 2 − 3i.
−2 + 3i, −3 − 2i, 2 − 3i
Problem 20 A real product
Expand: 6 − 15i + 2ki − 5ki2 = (6 + 5k) + (2k − 15)i.
The imaginary part is 0 when 2k = 15, so k = 7.5. The real part is 6 + 37.5 = 43.5.
k = 7.5 and the product is 43.5