🔢 Number Properties, Exponents and Roots

Part 1 of 7 — Number Properties on the ASVAB

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the fifth subtest on the computer version (CAT-ASVAB). You get 15 scored questions in 31 minutes, which works out to about 2 minutes per question. Every question has four answer choices, and there is no calculator. At MEPS you get two sheets of scratch paper and two pencils, and you can ask for more.

MK is one of the four subtests that build your AFQT score (AR + MK + 2·VE), the score the military uses to check whether you are eligible to enlist. It also feeds several job line scores, such as the Air Force A and E composites.

The official description of MK starts with "number theory" and "numeration." That is this topic: what kinds of numbers there are, how they behave, factors and primes, signs, exponents and roots. These questions are usually short. The challenge is knowing the rules cold so you can answer quickly and save time for the longer algebra and geometry questions.

Key Fact: The CAT is adaptive. A right answer usually brings a harder question next, and you can't go back to a question once you answer it. Fast, sure answers on number questions build momentum.

🔢 Number Properties, Exponents and Roots

Part 2 of 7 — Primes, Factors and Multiples

Factors and Multiples

A factor of a number divides it evenly. The factors of $18$ are $1, 2, 3, 6, 9, 18$.

A multiple of a number is that number times an integer. The multiples of $6$ are $6, 12, 18, 24, \dots$

Factors are small (no bigger than the number); multiples are big (at least the number). $6$ is a factor of $18$, and $18$ is a multiple of $6$.

Finding all factors fast: work in pairs from the outside in. For $36$: $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$. Stop when the pair meets. So $36$ has nine factors.

Prime and Composite

A prime number has exactly two factors: $1$ and itself. A composite number has more than two.

• The primes under $50$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$
• $1$ is not prime (it has only one factor).
• $2$ is the only even prime. Every other even number divides by $2$.

Divisibility Rules

Key Fact: To test whether a number is prime, try dividing by the primes $2, 3, 5, 7, \dots$ up to its square root. If none work, it is prime. For $91$: $91 = 7 \cdot 13$, so it is not prime, a favorite trap.

🔢 Number Properties, Exponents and Roots

Part 3 of 7 — Order of Operations, Integers and Absolute Value

Order of Operations (PEMDAS)

When an expression has more than one operation, do them in this order:

The trap: "MD" and "AS" are pairs. Multiplication does not always come before division; whichever comes first reading left to right goes first.

Worked Example 1

$18 - 12 \div 3 \cdot 2$

• Division and multiplication, left to right: $12 \div 3 = 4$, then $4 \cdot 2 = 8$
• Subtraction: $18 - 8 = 10$

(Multiplying first, $3 \cdot 2 = 6$, then $12 \div 6 = 2$, gives $16$. Wrong.)

Worked Example 2

$4 + 2(7 - 3)^2$

• Parentheses: $7 - 3 = 4$
• Exponent: $4^2 = 16$
• Multiply: $2 \cdot 16 = 32$
• Add: $4 + 32 = 36$

Worked Example 3: a fraction bar groups

$\frac{20 - 2 \cdot 4}{3 + 1} = \frac{20 - 8}{4} = \frac{12}{4} = 3$

Simplify the top and the bottom completely before you divide.

🔢 Number Properties, Exponents and Roots

Part 4 of 7 — Exponents and Scientific Notation

What an Exponent Means

An exponent says how many times to multiply the base by itself:

$3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$

It is not base times exponent: $3^4 \ne 12$.

Powers Worth Knowing by Heart

The Exponent Rules (same base)

The rules only work when the bases match. $2^3 \cdot 3^2$ is just $8 \cdot 9 = 72$; you can't add the exponents.

Negative Bases

Parentheses matter:

$(-3)^2 = (-3)(-3) = 9 \qquad -3^2 = -(3 \cdot 3) = -9$

Without parentheses the exponent applies only to the $3$. A negative base to an even power is positive; to an odd power it is negative: $(-2)^5 = -32$.

Key Fact: A negative exponent does not make the number negative. $4^{-2} = \frac{1}{16}$, a small positive number.

🔢 Number Properties, Exponents and Roots

Part 5 of 7 — Square Roots and Radicals

Square roots are named in the official description of MK, so expect them. Most root questions are quick if you know your perfect squares.

Perfect Squares to Memorize

The square root $\sqrt{x}$ is the nonnegative number whose square is $x$: $\sqrt{144} = 12$.

Cube Roots

$\sqrt[3]{x}$ is the number whose cube is $x$. Know $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$, $\sqrt[3]{64} = 4$, $\sqrt[3]{125} = 5$, $\sqrt[3]{1{,}000} = 10$. A cube root of a negative number is negative: $\sqrt[3]{-27} = -3$.

Estimating a Root

$\sqrt{55}$ is between $\sqrt{49} = 7$ and $\sqrt{64} = 8$, and closer to $7$ (since $55$ is closer to $49$). So $\sqrt{55} \approx 7.4$. On the ASVAB, estimating is often enough to pick the right choice.

Roots of Fractions and Decimals

$\sqrt{\frac{9}{25}} = \frac{3}{5} \qquad \sqrt{0.49} = 0.7 \qquad \sqrt{0.04} = 0.2$

Check decimals by squaring: $0.7 \times 0.7 = 0.49$. ✔

🔢 Number Properties, Exponents and Roots

Part 6 of 7 — Test Strategy and Common Traps

Your Clock

On the CAT-ASVAB, MK gives you 31 minutes for 15 questions, about 2 minutes each. (The paper version is tighter: 25 questions in 24 minutes, about 1 minute each.) Number-property questions are usually the fastest on the subtest. Aim to finish them in under a minute so you bank time for geometry and word-heavy algebra.

How the Wrong Answers Are Built

The three wrong choices are rarely random. Each one is the answer you get from a specific mistake. If you know the mistakes, you can spot the traps.

Five Habits That Save Points

1. Write every step on scratch paper. Sign errors happen in your head, not on paper.
2. Estimate first. If $\sqrt{130}$ must be a little more than $11$, any choice near $65$ is out.
3. Plug in small numbers for "which must be even/odd/positive" questions. Try $1$, $2$, $-1$, and $0$ if it is allowed.
4. Check the other direction. After finding a root, square it. After a factorization, multiply it back.
5. Answer every question genuinely. You can't go back on the CAT, and the official guidance is that a string of wrong answers at the end of a subtest carries a relatively large penalty, so keep working real answers even when time is short rather than clicking at random.

Key Fact: Some questions may be unscored tryouts, and you can't tell which. Treat every question as if it counts.

🔢 Number Properties, Exponents and Roots

Part 7 of 7 — Mixed Mastery Set

Twelve questions from every part of this lesson, in the mix you'll see on the real MK subtest: a few quick ones, a solid middle, and a few that take real care.

Test conditions: no calculator, scratch paper only, and aim for about 2 minutes per question. Work each one fully before you look at the explanation, just as you would on the CAT, where you can't go back.