MA2 Number & Quantity

Number & Quantity is the part of ACT Math that asks what a number is and how it behaves: exponents and radicals, scientific notation, absolute value, factors and multiples, remainders, complex numbers, matrices and vectors. Most of it is short computation. That is good news — these are the questions you can finish in under a minute once the rules are automatic, which buys you time for the long word problems later in the section.

The exponent rules, all of them, on one card

Half of this category is these seven lines. They only work when the bases match.

Rule	What it says	Example
Product	a^m · aⁿ = a^m+n	2³ · 2⁵ = 2⁸
Quotient	a^m ÷ aⁿ = a^m−n	x⁸ ÷ x⁵ = x³
Power of a power	(a^m)ⁿ = a^mn	(x⁴)³ = x¹²
Power of a product	(ab)ⁿ = aⁿbⁿ	(3x²)³ = 27x⁶
Zero	a⁰ = 1 (a ≠ 0)	7⁰ = 1
Negative	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8
Fractional	a^m/n = the n-th root of a^m	8^2/3 = 4

Two habits make these reliable. First, handle the coefficient and each variable separately — 12x⁸y³ ÷ 4x⁵y⁷ is 12 ÷ 4, then x⁸ ÷ x⁵, then y³ ÷ y⁷, three small problems instead of one big one. Second, in a^m/n take the root first: 64^2/3 is (∛64)² = 4² = 16, which you can do in your head, while 64² = 4,096 and then a cube root is a mess.

A negative exponent is a reciprocal, not a negative number. 2⁻³ = 1/2³ = 1/8. It is positive. The minus sign tells you which side of the fraction bar the factor belongs on — nothing else. So x⁻⁴ in a numerator means x⁴ in the denominator, and (2/3)⁻² = (3/2)² = 9/4.

Radicals: the one error that costs the most points

√(a + b) is not √a + √b. Test it once and you will never do it again: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The radical sign is a grouping symbol — everything under it gets combined first. The same goes for squaring: (a + b)² is not a² + b².

Simplify by pulling out perfect squares: √72 = √(36 · 2) = 6√2. You are done only when nothing square is left inside.
Multiplication and division do split: √a · √b = √(ab) and √a ÷ √b = √(a/b). Addition and subtraction do not.
Add like radicals like terms: 3√5 + 2√5 = 5√5. The radicand never changes. If the radicands look different, simplify first — 3√20 + √45 becomes 6√5 + 3√5 = 9√5.
Rationalise by multiplying top and bottom by the same thing: 12/√6 = 12√6/6 = 2√6. For a two-term denominator use the conjugate: 6/(3 − √5) = 6(3 + √5)/((3)² − 5) = (18 + 6√5)/4 = (9 + 3√5)/2.

Scientific notation

A number in scientific notation is a first factor at least 1 and less than 10, times a power of ten. Multiplying multiplies the first factors and adds the exponents; dividing divides the first factors and subtracts them. Then check the first factor and renormalise: 3.2 × 10⁵ times 4 × 10⁻² is 12.8 × 10³, which is not yet in scientific notation — move the decimal one place left and add one to the exponent to get 1.28 × 10⁴. For "how many times greater", divide the larger by the smaller and keep only the power of ten.

Powers of i cycle with period 4

i is defined by i² = −1. Everything else follows: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then it starts over. To find i⁴⁷, divide the exponent by 4 and keep the remainder: 47 = 4 · 11 + 3, so i⁴⁷ = i³ = −i. Complex numbers add like terms and multiply like binomials — the only extra step is replacing i² with −1 at the end. To divide, multiply the top and bottom by the conjugate of the denominator: (c + di)(c − di) = c² + d², a real number, which is exactly what clears i out of the bottom.

Matrices: read the dimensions before you do anything

A matrix is written rows first: an m × n matrix has m rows and n columns. In this module a matrix is written inline as [3 −1; 2 5], meaning the first row is 3 and −1, and the second row is 2 and 5. Write the two sizes side by side to decide whether a product exists:

(3 × 4)(4 × 2): the inner numbers match, so AB exists and is 3 × 2 — the outer numbers.
(2 × 3)(2 × 4): the inner numbers differ, so AB is not defined, no matter what the entries are.
Order matters. AB and BA are usually different, and one can exist while the other does not.
Adding and scalar multiplication work entry by entry, and two matrices can only be added when they are the same size. Multiplication does not work entry by entry: each entry of AB is a row of A paired with a column of B, multiplied term by term and added.
For a 2 × 2, the determinant of [a b; c d] is ad − bc.

The trap: "Which of the following is equivalent to √(x² + 16)?" There is no simplification — √(x² + 16) is already as simple as it gets, and x + 4 is a different number for every x except one. Whenever an answer choice splits a radical or a square across a sum, check it with actual numbers. Put x = 3: √(9 + 16) = 5, while 3 + 4 = 7. One substitution kills the wrong choice for good.

The rest of the category, briefly

Rational or irrational. A number is rational if it can be written as one integer over another. Fractions, terminating decimals and repeating decimals all qualify — so does √49, because it is 7. √50 does not, because 50 is not a perfect square.
GCF and LCM. Factor both numbers into primes. The greatest common factor takes each shared prime to the lower power; the least common multiple takes every prime that appears to the higher power. "Largest number of identical groups" is a GCF; "when do the two events next coincide" is an LCM.
Divisibility. By 2 if it is even; by 3 if the digits add to a multiple of 3; by 4 if the last two digits do; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and 3 tests; by 9 if the digits add to a multiple of 9.
Units digits and remainders. Powers repeat in short cycles — the units digits of 7ⁿ run 7, 9, 3, 1 and then repeat. Divide the exponent by the cycle length and use the remainder, remembering that a remainder of 0 means the last entry of the cycle, not the first.
Vectors. ⟨a, b⟩ adds component by component, and its magnitude is √(a² + b²) — the hypotenuse of a right triangle with legs a and b.
Unit analysis. Write every rate as a fraction and cancel units on paper. If the units left over are not the ones the question asked for, one of your factors is upside down.

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