Algebra 2 Practice Test: Operations with Complex Numbers
- Operations with Complex Numbers
- 10 questions
- 15 minutes
These 10 questions practice powers of i, the four operations with complex numbers, conjugates and square roots of negative numbers. Write every square root of a negative number as a multiple of i before you multiply.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the value of i50?
Correct answer: B
Powers of i repeat every 4 steps. 50 = 4 · 12 + 2, so i50 = i2 = −1.
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Question 2
Simplify (5 − 3i) + (−2 + 7i).
Correct answer: C
Add real parts and imaginary parts separately: 5 + (−2) = 3 and −3i + 7i = 4i. Subtracting −2 instead of adding it gives 7 + 4i.
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Question 3
Which expression is equivalent to (2 − i)2?
Correct answer: A
(2 − i)2 = 4 − 4i + i2 = 4 − 4i − 1 = 3 − 4i. Using i2 = 1 gives 5 − 4i, and squaring each part separately loses the middle term.
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Question 4
Which expression is equivalent to (1 + 2i)(3 − i)?
Correct answer: B
3 − i + 6i − 2i2 = 3 + 5i + 2 = 5 + 5i. Treating −2i2 as −2 gives 1 + 5i.
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Question 5
Which expression is equivalent to 103 + i?
Correct answer: D
Multiply by the conjugate: 10(3 − i)(3 + i)(3 − i) = 30 − 10i9 + 1 = 3 − i. Computing the denominator as 9 − 1 = 8 gives the fraction with denominator 4.
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Question 6
What is the value of −9 · −4?
Correct answer: C
−9 = 3i and −4 = 2i, so the product is 6i2 = −6. The rule a · b = ab does not hold for two negative numbers, which is why 6 is wrong.
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Question 7
What is the complex conjugate of −4 + 7i?
Correct answer: B
The conjugate keeps the real part and changes the sign of the imaginary part: −4 + 7i becomes −4 − 7i.
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Question 8
For real numbers x and y, (x + 3) + 2yi = 7 − 8i. What are x and y?
Correct answer: D
Two complex numbers are equal when their real parts and imaginary parts are equal: x + 3 = 7 gives x = 4, and 2y = −8 gives y = −4.
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Question 9
What is the value of i + i2 + i3 + i4?
Correct answer: A
i + (−1) + (−i) + 1 = 0. Any four consecutive powers of i add up to 0.
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Question 10
Which expression is equivalent to 4 + 3i2i?
Correct answer: A
Multiply numerator and denominator by −i: (4 + 3i)(−i)2i(−i) = −4i + 32 = 3 − 4i2. Multiplying by i instead of −i flips both signs.
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