12 multiple-choice questions, progressively harder.
Simplify (7−3i)−(2−8i)(7 - 3i) - (2 - 8i)(7−3i)−(2−8i).
Solution
Correct answer: D
Distribute the minus sign to both parts of the second number, then combine.
(7−3i)−(2−8i)=(7−2)+(−3+8)i=5+5i(7 - 3i) - (2 - 8i) = (7 - 2) + (-3 + 8)i = 5 + 5i(7−3i)−(2−8i)=(7−2)+(−3+8)i=5+5i
The imaginary part is −3−(−8)=−3+8=5-3 - (-8) = -3 + 8 = 5−3−(−8)=−3+8=5.
Simplify (3+i)2(3 + i)^2(3+i)2.
Correct answer: A
Use the square-of-a-binomial pattern, then replace i2i^2i2 with −1-1−1.
(3+i)2=9+6i+i2=9+6i−1=8+6i(3 + i)^2 = 9 + 6i + i^2 = 9 + 6i - 1 = 8 + 6i(3+i)2=9+6i+i2=9+6i−1=8+6i
The i2i^2i2 becomes −1-1−1, lowering the real part from 999 to 888.
Multiply (7+3i)(7−3i)(7 + 3i)(7 - 3i)(7+3i)(7−3i).
A number times its conjugate gives a2+b2a^2 + b^2a2+b2.
(7+3i)(7−3i)=72+32=49+9=58(7 + 3i)(7 - 3i) = 7^2 + 3^2 = 49 + 9 = 58(7+3i)(7−3i)=72+32=49+9=58
Getting 49−9=4049 - 9 = 4049−9=40 uses the real-number difference of squares; here i2=−1i^2 = -1i2=−1 makes it a sum.
Simplify (4+9i)+(4−9i)(4 + 9i) + (4 - 9i)(4+9i)+(4−9i).
Correct answer: B
Adding a number to its conjugate cancels the imaginary part.
(4+9i)+(4−9i)=(4+4)+(9−9)i=8(4 + 9i) + (4 - 9i) = (4 + 4) + (9 - 9)i = 8(4+9i)+(4−9i)=(4+4)+(9−9)i=8
The imaginary parts 999 and −9-9−9 sum to zero, leaving the real number 888.
Simplify (2−3i)2(2 - 3i)^2(2−3i)2.
(2−3i)2=4−12i+9i2=4−12i−9=−5−12i(2 - 3i)^2 = 4 - 12i + 9i^2 = 4 - 12i - 9 = -5 - 12i(2−3i)2=4−12i+9i2=4−12i−9=−5−12i
The term 9i29i^29i2 becomes −9-9−9, so the real part is 4−9=−54 - 9 = -54−9=−5.
Multiply (−1+2i)(3+4i)(-1 + 2i)(3 + 4i)(−1+2i)(3+4i).
Correct answer: C
Expand term by term, then replace i2i^2i2 with −1-1−1.
(−1+2i)(3+4i)=−3−4i+6i+8i2=−3+2i−8=−11+2i(-1 + 2i)(3 + 4i) = -3 - 4i + 6i + 8i^2 = -3 + 2i - 8 = -11 + 2i(−1+2i)(3+4i)=−3−4i+6i+8i2=−3+2i−8=−11+2i
The term 8i28i^28i2 becomes −8-8−8, so the real part is −3−8=−11-3 - 8 = -11−3−8=−11.
Simplify (12+5i)−(7−9i)−(2+i)(12 + 5i) - (7 - 9i) - (2 + i)(12+5i)−(7−9i)−(2+i).
Distribute both minus signs, then combine the parts separately.
(12−7−2)+(5+9−1)i=3+13i(12 - 7 - 2) + (5 + 9 - 1)i = 3 + 13i(12−7−2)+(5+9−1)i=3+13i
Subtracting 7−9i7 - 9i7−9i adds 9i9i9i, so the imaginary part is 5+9−1=135 + 9 - 1 = 135+9−1=13.
Simplify (1+i)2(1 + i)^2(1+i)2.
(1+i)2=1+2i+i2=1+2i−1=2i(1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i(1+i)2=1+2i+i2=1+2i−1=2i
The real parts 111 and −1-1−1 cancel, leaving the pure imaginary 2i2i2i.
Multiply (5+i)(5−i)(5 + i)(5 - i)(5+i)(5−i).
(5+i)(5−i)=52+12=25+1=26(5 + i)(5 - i) = 5^2 + 1^2 = 25 + 1 = 26(5+i)(5−i)=52+12=25+1=26
Getting 25−1=2425 - 1 = 2425−1=24 mistakes the sum for the real-number difference of squares.
Multiply (4+i)(4+i)(4 + i)(4 + i)(4+i)(4+i).
This is the square (4+i)2(4 + i)^2(4+i)2. Use the binomial pattern, then replace i2i^2i2 with −1-1−1.
(4+i)2=16+8i+i2=16+8i−1=15+8i(4 + i)^2 = 16 + 8i + i^2 = 16 + 8i - 1 = 15 + 8i(4+i)2=16+8i+i2=16+8i−1=15+8i
The i2i^2i2 becomes −1-1−1, lowering the real part from 161616 to 151515.
Simplify (10−4i)−(10+4i)(10 - 4i) - (10 + 4i)(10−4i)−(10+4i).
Subtract the parts separately, distributing the minus sign.
(10−4i)−(10+4i)=(10−10)+(−4−4)i=−8i(10 - 4i) - (10 + 4i) = (10 - 10) + (-4 - 4)i = -8i(10−4i)−(10+4i)=(10−10)+(−4−4)i=−8i
The real parts cancel and the imaginary parts give −4−4=−8-4 - 4 = -8−4−4=−8.
The product (6−5i)(6+5i)(6 - 5i)(6 + 5i)(6−5i)(6+5i) equals which real number?
(6−5i)(6+5i)=62+52=36+25=61(6 - 5i)(6 + 5i) = 6^2 + 5^2 = 36 + 25 = 61(6−5i)(6+5i)=62+52=36+25=61
Getting 36−25=1136 - 25 = 1136−25=11 forgets that i2=−1i^2 = -1i2=−1 turns the subtracted square into an added one.
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