Exponents and Square Roots: ASVAB Math Knowledge Rules
The exponent rules worth knowing for ASVAB Mathematics Knowledge, the negative-sign trap, and how to estimate square roots without a calculator.
Exponents and roots are a staple of high school math — the kind of material the official ASVAB program means by Mathematics Knowledge, "knowledge of high school mathematics principles." They're also a place where small slips cost easy points. Here are the rules, the classic trap, and how to handle roots without a calculator.
What an exponent means
An exponent says how many times to multiply a number by itself:
2⁵ = 2 × 2 × 2 × 2 × 2 = 32 3⁴ = 3 × 3 × 3 × 3 = 81
A common slip is multiplying the base by the exponent: 2⁵ is not 2 × 5 = 10.
The rules worth knowing
| Rule | In symbols | Example |
|---|---|---|
| Multiply same base: add exponents | xᵃ · xᵇ = xᵃ⁺ᵇ | 2³ · 2⁴ = 2⁷ = 128 |
| Divide same base: subtract exponents | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power: multiply exponents | (xᵃ)ᵇ = xᵃᵇ | (2³)² = 2⁶ = 64 |
| Zero exponent | x⁰ = 1 (x ≠ 0) | 5⁰ = 1 |
| Negative exponent: reciprocal | x⁻ᵃ = 1 / xᵃ | 2⁻³ = 1/8 |
The rules only combine exponents when the bases are the same. 2³ · 3² can't be simplified with a rule — you just calculate: 8 × 9 = 72.
The negative-sign trap
These two look almost identical and give opposite answers:
(−3)² = (−3) × (−3) = 9 −3² = −(3 × 3) = −9
Without parentheses, the exponent applies only to the 3, and the minus sign is applied afterward. On a multiple-choice test, both 9 and −9 can easily be among the choices — read the parentheses.
Square roots
A square root undoes a square: √144 = 12, because 12² = 144. Knowing the perfect squares up to 15² = 225 covers most of what you need:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cube roots work the same way with cubes: ∛27 = 3 (3 × 3 × 3 = 27) and ∛64 = 4.
Simplifying a root
Split out a perfect square:
√50 = √(25 × 2) = 5√2
Since √2 is about 1.414, 5√2 is about 7.07.
Estimating a root
When the number isn't a perfect square, place it between two that are:
√70: 64 < 70 < 81, so √70 is between 8 and 9 — and closer to 8 (it's about 8.37).
On a multiple-choice question, "between 8 and 9" is often all you need to pick the answer.
Roots of decimals
√0.25 = 0.5, because 0.5 × 0.5 = 0.25.
Notice the root is bigger than the number. That's normal for numbers between 0 and 1 — and a common reason people reject the right answer.
A quick self-check
- 2³ · 2² = ? → 32 (2⁵)
- (−2)⁴ = ? → 16
- 10⁰ = ? → 1
- √196 = ? → 14
- 3⁻² = ? → 1/9
Practice it
Our free practice test includes Mathematics Knowledge with the computer version's time limits — 15 questions in 31 minutes in the full version, no calculator — plus the other three AFQT sections, and ends with an AFQT estimate (a practice estimate, not an official score).
Related: The Pythagorean Theorem on the ASVAB uses squares and roots in every problem, and Mathematics Knowledge Practice Test covers the whole section.
Quick answers
What is any number to the power of zero?
1, for any number except 0.
What does a negative exponent mean?
Take the reciprocal: 2⁻³ = 1/2³ = 1/8.
Is −3² equal to 9 or −9?
−9. Without parentheses, the exponent applies only to the 3. (−3)² is 9.
How do I estimate a square root without a calculator?
Find the two perfect squares around it. √70 is between √64 = 8 and √81 = 9.
Sources
Link copied
Math
MathNew
MathNew