Number Properties, Exponents and Roots

🔢 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.

Warm-up. 🎯 No calculator. See what you already know; the rest of the lesson fills the gaps.

The Kinds of Numbers

The ASVAB uses these names without explaining them, so learn them now.

Every integer is rational ($-5 = \frac{-5}{1}$). A decimal that ends ($0.25$) or repeats ($0.333\dots$) is rational. The square root of a number that is not a perfect square, such as $\sqrt{2}$, is irrational.

Even and Odd

An even number divides evenly by $2$; an odd number does not. Zero is even. Negative numbers can be even or odd too ($-4$ is even, $-7$ is odd).

You don't have to memorize the table: test it with small numbers. Odd + odd? $3 + 5 = 8$, even.

The Properties That Make Mental Math Easy

Subtraction and division are not commutative: $10 - 4 \ne 4 - 10$ and $12 \div 3 \ne 3 \div 12$.

Worked Example: the distributive property as a shortcut

Find $7 \cdot 98$ without a calculator.

$7 \cdot 98 = 7(100 - 2) = 700 - 14 = 686$

Worked Example: grouping to make tens

Find $25 \cdot 13 \cdot 4$.

Regroup: $(25 \cdot 4) \cdot 13 = 100 \cdot 13 = 1{,}300$. The commutative and associative properties let you multiply in whatever order is easiest.

Reciprocals

The reciprocal of a number is $1$ divided by it: flip the fraction. The reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$; the reciprocal of $6$ is $\frac{1}{6}$. A number times its reciprocal is always $1$. Zero has no reciprocal.

Name the number, name the property. 🎯

Number vocabulary. 🔽 For each line, choose the most specific true description or the correct value.

Mental math with the properties. ✍️ No calculator.

1) $9 \cdot 102$ (use $9(100 + 2)$):

2) $50 \cdot 17 \cdot 2$ (regroup first):

3) $\frac{5}{6}$ times its reciprocal:

Key Takeaways — Part 1

• MK on the CAT-ASVAB: 15 questions, 31 minutes, about 2 minutes each, four choices, no calculator
• Whole numbers start at $0$; integers add the negatives; rational numbers are fractions of integers; irrational numbers ($\pi$, $\sqrt{2}$) are not
• Zero is even; odd + odd is even; odd × odd is odd; even × anything is even
• Distributive: $a(b + c) = ab + ac$, a powerful mental-math shortcut
• A number times its reciprocal is $1$

Up Next: Part 2 — primes, factors, multiples, the GCF and the LCM.

🔢 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.

Prime Factorization, GCF and LCM

Prime factorization

Break a number into primes with a factor tree. For $60$:

$60 = 6 \cdot 10 = (2 \cdot 3)(2 \cdot 5) = 2^2 \cdot 3 \cdot 5$

Every composite number has exactly one prime factorization.

Greatest Common Factor (GCF)

The GCF is the largest number that divides both numbers. Use the primes they share, with the smaller power.

Example: GCF of $60$ and $84$.

• $60 = 2^2 \cdot 3 \cdot 5$
• $84 = 2^2 \cdot 3 \cdot 7$
• Shared: $2^2 \cdot 3 = 12$

Least Common Multiple (LCM)

The LCM is the smallest number both divide into. Use every prime that appears, with the larger power.

Example: LCM of $12$ and $18$.

• $12 = 2^2 \cdot 3$
• $18 = 2 \cdot 3^2$
• All primes, biggest powers: $2^2 \cdot 3^2 = 36$

Quick method for small numbers: list multiples of the larger number until the smaller one divides it. Multiples of $18$: $18$ (no), $36$ (yes). LCM $= 36$.

Which one does the word problem want?

Example (GCF): A supply sergeant has $24$ canteens and $36$ first-aid kits and wants identical packs using every item. The most packs possible is the GCF of $24$ and $36$, which is $12$ (each pack gets $2$ canteens and $3$ kits).

Example (LCM): One generator is checked every $4$ days and another every $6$ days. Both are checked today. They are next checked on the same day in LCM$(4, 6) = 12$ days.

Primes and divisibility. 🎯

GCF or LCM? 🎯 Decide which the problem needs, then find it.

Factors of 48. 🔽 Work in pairs from the outside in: $1 \times 48$, $2 \times 24$, …

Find each value. ✍️

1) The GCF of $54$ and $72$:

2) The LCM of $8$ and $14$:

3) A unit has $30$ rifles and $45$ helmets to sort into identical crates, using every item. The greatest number of crates possible:

Key Takeaways — Part 2

• Factors divide a number; multiples are the number times an integer
• Prime = exactly two factors. $1$ is not prime; $2$ is the only even prime; watch for $51$, $57$, $91$
• Divisibility: $3$ and $9$ use the digit sum; $4$ uses the last two digits
• GCF: shared primes, smaller powers. Equal groups, nothing left over
• LCM: all primes, larger powers. Cycles that line up again

Up Next: Part 3 — order of operations, negative numbers and absolute value.

🔢 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.

Negative Numbers

Adding and subtracting

• Same signs: add the sizes, keep the sign. $-6 + (-9) = -15$
• Different signs: subtract the sizes, keep the sign of the bigger size. $-14 + 5 = -9$
• Subtracting a negative is adding: $7 - (-3) = 7 + 3 = 10$

Think of a thermometer: $-4^\circ$ rising $10$ degrees is $6^\circ$, so $-4 + 10 = 6$.

Multiplying and dividing

Same rule for division. With several factors, count the negatives: an even count gives a positive answer, an odd count gives a negative one.

$(-2)(-3)(-5) = -30 \quad \text{(three negatives: negative)}$

Absolute Value

The absolute value of a number is its distance from $0$ on the number line, so it is never negative.

$\lvert -8 \rvert = 8 \qquad \lvert 8 \rvert = 8 \qquad \lvert 0 \rvert = 0$

Absolute value bars are grouping symbols: work inside first, then take the absolute value.

$\lvert 3 - 11 \rvert = \lvert -8 \rvert = 8 \qquad -\lvert -5 \rvert = -5$

In the second example the negative sign is outside the bars, so it stays.

Key Fact: $\lvert a - b \rvert$ is the distance between $a$ and $b$ on the number line. The distance from $-3$ to $9$ is $\lvert 9 - (-3) \rvert = 12$.

Order of operations. 🎯 No calculator.

Signs and absolute value. 🎯

Step through $40 - 3(2 - 6)^2$. 🔽

Compute. ✍️

1) $-7 + 15 - 20$:

2) $(-48) \div (-6) - 10$:

3) The distance on a number line between $-11$ and $4$:

Key Takeaways — Part 3

• PEMDAS, but multiplication/division and addition/subtraction each go left to right
• Fraction bars and absolute value bars group: simplify inside first
• Subtracting a negative is adding
• Count the negatives: even count → positive product, odd count → negative
• Absolute value is distance from $0$: never negative; $\lvert a - b \rvert$ is the distance between $a$ and $b$

Up Next: Part 4 — exponents, their rules and scientific notation.

🔢 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.

Exponent rules. 🎯 No calculator.

Scientific Notation

Scientific notation writes a number as

$(\text{a number from } 1 \text{ up to but not including } 10) \times 10^{n}$

Big numbers: positive exponent

$4{,}700{,}000 = 4.7 \times 10^6$

Count how many places the decimal point moves to get $4.7$: six places to the left, so $10^6$.

Small numbers: negative exponent

$0.00032 = 3.2 \times 10^{-4}$

The point moves four places to the right to get $3.2$, so the exponent is $-4$.

Converting back

• $6.05 \times 10^3$: move the point $3$ places right → $6{,}050$
• $9 \times 10^{-2}$: move the point $2$ places left → $0.09$

Multiplying and dividing

Handle the front numbers and the powers of ten separately.

$(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$

$\frac{8 \times 10^7}{4 \times 10^2} = 2 \times 10^5$

If the front number ends up $10$ or more, adjust:

$(5 \times 10^3)(4 \times 10^2) = 20 \times 10^5 = 2 \times 10^6$

Key Fact: $0.5 \times 10^4$ and $25 \times 10^3$ are not in scientific notation. The front number must be at least $1$ and less than $10$.

Scientific notation. 🔽

Compute. ✍️ Fractions are fine (type $1/8$ as 1/8).

1) $3^2 \cdot 3^1$:

2) $10^{-3}$ as a decimal:

3) $(-1)^{7} + 6^0$:

4) $\dfrac{9 \times 10^8}{3 \times 10^5} = 3 \times 10^{n}$. What is $n$?

Key Takeaways — Part 4

• $a^n$ means multiply $a$ by itself $n$ times: $2^5 = 32$, not $10$
• Same base: multiply → add exponents; divide → subtract; power of a power → multiply
• $a^0 = 1$; $a^{-n} = \frac{1}{a^n}$ (still positive)
• $(-3)^2 = 9$ but $-3^2 = -9$
• Scientific notation: front number from $1$ to under $10$; big numbers → positive exponent, small → negative

Up Next: Part 5 — square roots, cube roots and simplifying radicals.

🔢 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: $0.7 \times 0.7 = 0.49$. ✔

Roots without a calculator. 🎯

Simplifying Radicals

Answer choices often show roots in simplest form, like $5\sqrt{3}$ instead of $\sqrt{75}$.

Method: find the largest perfect square factor, and take its root out front.

$\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$

$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

If you pick a smaller square, you just have to keep going: $\sqrt{72} = \sqrt{4 \cdot 18} = 2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}$.

Multiplying and dividing roots

$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

• $\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6$
• $\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$
• $(2\sqrt{5})(3\sqrt{5}) = 6 \cdot 5 = 30$

Adding and subtracting roots

Only like radicals (same number under the root) combine, the way like terms do:

$4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$

$\sqrt{2} + \sqrt{3}$ does not simplify. Sometimes simplifying first creates like radicals:

$\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$

Key Fact: Roots do not split over addition. $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 7$. The wrong one is almost always an answer choice.

Simplify. 🎯

Fill in the number. ✍️

1) $\sqrt{98} = a\sqrt{2}$. What is $a$?

2) $\sqrt{0.81}$ as a decimal:

3) $\sqrt{36 + 64}$:

4) $(5\sqrt{2})^2$:

Key Takeaways — Part 5

• Know the perfect squares through $15^2 = 225$, plus the cubes $8, 27, 64, 125$
• Estimate a root between the two perfect squares around it
• Simplify with the largest perfect square factor: $\sqrt{75} = 5\sqrt{3}$
• $\sqrt{a}\sqrt{b} = \sqrt{ab}$; only like radicals add
• $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$

Up Next: Part 6 — how these questions are built to trap you, and how to move fast.

🔢 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.

Spot the trap. 🎯 Each question asks which mistake produces a choice.

Work smart, not long. 🎯 Use estimating or plugging in.

Match each wrong answer to its mistake. 🔽 The question was: evaluate $(-3)^2 + 2^3$. The correct answer is $17$.

Key Takeaways — Part 6

• About 2 minutes per MK question on the CAT; number questions should take less
• Wrong choices come from predictable mistakes: left-to-right only, sign slips, base × exponent, GCF/LCM swaps, splitting roots
• Estimate to eliminate, plug in small numbers for "must be" questions, check backward
• No going back on the CAT: answer each question for real and keep a steady pace

Up Next: Part 7 — a mixed mastery set at ASVAB difficulty.

🔢 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.

Set A. 🎯 Warm-up level.

Set B. 🎯 Typical MK level.

Set C. 🎯 The harder end of the subtest.

Last one. ✍️

1) What is the smallest positive whole number that is divisible by $2$, $3$, $4$ and $5$?

Key Takeaways — Number Properties, Exponents and Roots

• Number types: whole, integer, rational, irrational; $0$ is even; $2$ is the only even prime
• GCF = shared primes, smaller powers; LCM = all primes, larger powers
• PEMDAS with left-to-right pairs; fraction bars and absolute value bars group
• Signs: subtracting a negative adds; count the negatives in a product
• Exponents: add when multiplying, subtract when dividing, multiply for a power of a power; $a^0 = 1$; $a^{-n} = \frac{1}{a^n}$
• Scientific notation: front number from $1$ to under $10$
• Roots: largest perfect square factor; only like radicals add; $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$

Next: take the exit quiz for this topic, then move on to Linear Equations and Inequalities.