Question 1Easier
Given that is the imaginary unit, which of the following numbers is equal to ?
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Answer: C —
Expanding gives . Since , this becomes .
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Complex numbers — ACT practice questions
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Practice 46 Complex numbers questions in the app
What this topic is
On the ACT Mathematics test, Complex numbers covers products of a + bi numbers, powers of the imaginary unit i, and squares of complex binomials.
A student must multiply complexes, reduce high powers of i from i squared equaling negative one, and expand a real-minus-imaginary square. Items often ask which statement cannot be true of a given product, which listed number equals a squared complex quantity, or the value of a difference of two powers of i.
Traps include the wrong remainder when reducing an exponent and dropping a sign after replacing i squared. Choices are typically other a + bi numbers or competing claims about the factors.
Sample questions
Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.
Question 2Mid
For complex numbers and , . Which of the following statements must be true?
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Answer: D — At least one of and is outside the set of rational numbers.
The product of two rational numbers must be rational, but is irrational. Each other condition is possible with suitable factors.
Question 3Easier
Given that , what is the value of ?
Show answer
Answer: D —
Powers of repeat every 4 powers, so and . Therefore, .
Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.
More ACT topics
Mathematics
- Real number system and number properties
- Integer and rational exponents
- Radicals and roots
- Vectors and matrices
- Fractions, decimals, and operations
- Percents and percent change
- Ratios, rates, and proportional relationships
- Linear equations and expressions in one variable
- Inequalities (linear and compound)
- Systems of equations
- Polynomial expressions and factoring
- Quadratic equations
- Radical and rational equations
- Exponential relationships
- Function notation and evaluating functions
- Linear functions and graphs
- Quadratic and polynomial functions
- Radical and rational functions
- Piecewise functions
- Exponential and logarithmic functions
- Key features and transformations of graphs
- Lines, angles, and polygons
- Triangles — congruence, similarity, Pythagorean theorem
- Circles — arcs, sectors, angles, equations
- Surface area and volume of solids
- Coordinate geometry — distance, midpoint, slope, line equations
- Conic sections
- Right-triangle trigonometry and trig functions
- Measures of center and spread
- Data collection methods and study design
- Bivariate data and relationships (scatterplots, correlation)
- Probability and sample spaces
- Counting, combinations, and permutations
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