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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What this topic is
The imaginary unit i is defined so that i squared equals -1, and that single fact drives most Complex numbers questions.
Powers of i repeat in a cycle of four: i, -1, -i, 1, so a large power can be reduced by finding its remainder after division by four. Squaring a number such as 2 + i requires the full expansion, which gives 4 + 4i + i squared, or 3 + 4i.
ACT Mathematics asks for simplified sums, products and powers, and answer choices usually differ by a sign or by whether i squared was replaced with -1.
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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 ?
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Answer: D —
Powers of repeat every 4 powers, so and . Therefore, .
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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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