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ACT · Domain 1

Number & Quantity

7–10% of the ACT math section

Real and complex numbers, exponents and radicals, and — unlike the SAT — vectors and matrices. A small share of the test, and the questions are short: two rules applied cleanly, or a mistake in the first line.

Number & Quantity · Topic 1 of 3

Exponents & Radicals

Two rules do most of the work: a power of a product applies to every factor, and multiplying like bases adds exponents. Radicals are exponents in another notation — x\sqrt{x} is x1/2x^{1/2} — and combine only once they share a radicand.

The problem
For x≠0x \ne 0, which of the following is equivalent to (2x3)2⋅x−4(2x^3)^2 \cdot x^{-4}?
  1. A4x24x^2
  2. B2x22x^2
  3. C4x104x^{10}
  4. D4x−124x^{-12}
Working1 / 3
  1. The square applies to the 2 and to the x3x^3(2x3)2=4x6(2x^3)^2 = 4x^6
Number & Quantity · Topic 2 of 3

Complex Numbers, Vectors & Matrices

The ACT asks for arithmetic on complex numbers, a vector added to another or scaled, and the odd matrix added or multiplied by a number. None of it is deep; all of it uses one fact each. For complex numbers the fact is i2=−1i^2 = -1, and multiplication is FOIL with that substituted at the end.

The problem
What is (3+2i)(1−i)(3 + 2i)(1 - i)?
  1. F5−i5 - i
  2. G1−i1 - i
  3. H3−i3 - i
  4. J5+i5 + i
Working1 / 3
  1. FOIL3−3i+2i−2i23 - 3i + 2i - 2i^2
Number & Quantity · Topic 3 of 3

Number Properties & Scientific Notation

Primes, factors and multiples, absolute value, and numbers written as a digit times a power of ten. The ACT asks these fast and early. The one that costs time is the least common multiple, and it only costs time when it is done by listing — factor into primes and take each prime’s highest power.

The problem
The least common multiple of 12 and 18 is how much greater than their greatest common factor?
  1. A3030
  2. B3636
  3. C66
  4. D216216
Working1 / 4
  1. Factor both into primes12=22⋅3,18=2⋅3212 = 2^2 \cdot 3, \quad 18 = 2 \cdot 3^2