ASVAB exponents, roots and scientific notation
Exponent questions on ASVAB Mathematics Knowledge follow six rules: add exponents when you multiply, subtract when you divide, multiply when you raise a power to a power, anything to the zero power is 1, a negative exponent means a reciprocal, and roots undo powers. Scientific notation uses the same rules with powers of 10.
What exponent rules do you need for the ASVAB?
You need the rules in this table. They all work only when the bases are the same. Mathematics Knowledge covers "knowledge of high school mathematics principles" (What to Expect), and these rules are standard algebra.
| Rule | Example |
|---|---|
| xᵃ × xᵇ = xᵃ⁺ᵇ | 2³ × 2⁴ = 2⁷ = 128 |
| xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 5⁶ ÷ 5⁴ = 5² = 25 |
| (xᵃ)ᵇ = xᵃᵇ | (3²)³ = 3⁶ = 729 |
| (xy)ᵃ = xᵃyᵃ | (2x)³ = 8x³ |
| x⁰ = 1 | 7⁰ = 1 |
| x⁻ᵃ = 1/xᵃ | 2⁻³ = 1/8 |
| x^(1/2) = √x | 9^(1/2) = 3 |
Example: Simplify 2³ × 2⁴.
- Same base, multiplying: add the exponents. 3 + 4 = 7.
- 2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128.
Answer: 2⁷ = 128
Example: Simplify (3²)³.
- Power of a power: multiply the exponents. 2 × 3 = 6.
- 3⁶ = 9 × 9 × 9 = 81 × 9 = 729.
Answer: 3⁶ = 729
A common trap is adding exponents when the bases differ. 2³ × 3² is 8 × 9 = 72. It is not 6⁵.
How do zero and negative exponents work?
Any nonzero number to the zero power is 1. A negative exponent flips the number into a fraction: x⁻ᵃ = 1/xᵃ. A negative exponent does not make the answer negative.
Example: Evaluate 5⁰ and 2⁻³.
- 5⁰ = 1, because any nonzero base to the 0 power is 1.
- 2⁻³ = 1/2³ = 1/8 = 0.125.
Answer: 1 and 1/8
Watch the parentheses with negative bases. (−2)⁴ = 16 because four negatives multiply to a positive. Without parentheses, −2⁴ means −(2⁴) = −16. An even power of a negative number is positive; an odd power is negative.
How do you simplify square roots?
Pull out the largest perfect square factor. √72 = √(36 × 2) = 6√2. To do this quickly you need the perfect squares memorized.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
Example: Simplify √50 + √18.
- √50 = √(25 × 2) = 5√2.
- √18 = √(9 × 2) = 3√2.
- Like radicals add like terms: 5√2 + 3√2 = 8√2.
Answer: 8√2
You can only add roots when the number under the root is the same after simplifying. √2 + √3 does not combine. Multiplying is different: √a × √b = √(ab), so √3 × √12 = √36 = 6.
How do you estimate a square root without a calculator?
Find the two perfect squares on either side. √90 is between √81 = 9 and √100 = 10, and closer to 9 because 90 is nearer to 81. Its value is about 9.49.
Estimation is enough for most root questions, because the choices usually differ by more than the error. Cube roots work the same way with the cubes: 1, 8, 27, 64, 125.
How does scientific notation work?
A number in scientific notation is written as a × 10ⁿ where a is at least 1 and less than 10. The exponent n counts how many places the decimal point moves: positive for big numbers, negative for small ones.
| Standard form | Scientific notation |
|---|---|
| 45,000 | 4.5 × 10⁴ |
| 3,600,000 | 3.6 × 10⁶ |
| 0.00045 | 4.5 × 10⁻⁴ |
| 0.072 | 7.2 × 10⁻² |
Example: Write 0.00045 in scientific notation.
- Move the decimal right until one nonzero digit is in front: 4.5.
- You moved it 4 places to the right, so the exponent is −4.
Answer: 4.5 × 10⁻⁴
Example: Multiply (4.2 × 10⁵)(3 × 10⁻²).
- Multiply the front numbers: 4.2 × 3 = 12.6.
- Add the exponents: 5 + (−2) = 3. So 12.6 × 10³.
- 12.6 is not between 1 and 10, so shift: 1.26 × 10⁴.
- Check: 12.6 × 1,000 = 12,600 = 1.26 × 10⁴.
Answer: 1.26 × 10⁴ (12,600)
How do you simplify expressions with variables and exponents?
Handle the numbers and each variable separately. Multiply or divide the coefficients as usual, then apply the exponent rules to each letter on its own.
Example: Simplify (3x²y)(4xy³).
- Numbers: 3 × 4 = 12.
- x terms: x² × x¹ = x³ (a letter with no exponent has exponent 1).
- y terms: y¹ × y³ = y⁴.
Answer: 12x³y⁴
Example: Simplify 6x⁵ ÷ 2x².
- Numbers: 6 ÷ 2 = 3.
- x terms: x⁵ ÷ x² = x³.
Answer: 3x³
The step people skip is giving a bare variable its exponent of 1. In the first example, forgetting that x means x¹ leads to x² instead of x³. Write the 1 in small if it helps you.
Where do exponents fit in the order of operations?
Exponents come right after parentheses and before multiplication, division, addition and subtraction. The usual memory aid is PEMDAS: parentheses, exponents, multiplication and division from left to right, then addition and subtraction from left to right.
Example: Evaluate 3 + 2 × 4².
- Exponent first: 4² = 16.
- Then multiply: 2 × 16 = 32.
- Then add: 3 + 32 = 35.
Answer: 35
Doing the operations left to right gives (3 + 2) × 4² = 80 or (3 + 2 × 4)² = 121, both wrong. Another common trap is spreading an exponent over a sum. (3 + 2)² = 5² = 25, but 3² + 2² = 9 + 4 = 13. The exponent applies to the whole parentheses, so add first, then square.
How do you divide in scientific notation and take roots of decimals?
To divide, divide the front numbers and subtract the exponents. Then adjust so the front number is between 1 and 10.
Example: Divide (8 × 10⁶) by (2 × 10²).
- Fronts: 8 ÷ 2 = 4.
- Exponents: 6 − 2 = 4.
Answer: 4 × 10⁴ (40,000)
Example: Divide (1.5 × 10³) by (5 × 10⁵).
- Fronts: 1.5 ÷ 5 = 0.3.
- Exponents: 3 − 5 = −2. So 0.3 × 10⁻².
- 0.3 is below 1, so write it as 3 × 10⁻¹ and add the exponents: 3 × 10⁻³.
Answer: 3 × 10⁻³ (0.003)
Square roots of fractions and decimals use the same perfect squares. The square root of a fraction is the root of the top over the root of the bottom: √(16/25) = 4/5. For decimals, count places: √0.09 = 0.3 because 0.3 × 0.3 = 0.09, and √0.49 = 0.7. The root has half as many decimal places as the number under the root.
How do you avoid exponent mistakes on test day?
Slow down at the step where you choose a rule. Most wrong answers on these items are the result of using the right arithmetic with the wrong rule.
- Bases must match before you add or subtract exponents.
- Multiply exponents only for a power of a power, like (x²)³.
- A negative exponent gives a fraction, not a negative number.
- In scientific notation, make sure the front number is between 1 and 10 at the end.
Exponents also show up inside geometry (squared units, the Pythagorean theorem) and algebra. Review the algebra lesson and the geometry review, then practice on the Mathematics Knowledge page.
Practice questions
Work each one on paper first, then open the explanation. These are original AptiStep items, not real ASVAB questions.
Simplify x⁵ × x³.
- x¹⁵
- x⁸
- x²
- 2x⁸
Show answer and explanation
Answer: B. x⁸
When you multiply powers with the same base, add the exponents: 5 + 3 = 8, so the answer is x⁸. Multiplying the exponents (x¹⁵) is the rule for a power of a power, not for a product.
What is (2³)²?
- 32
- 12
- 64
- 36
Show answer and explanation
Answer: C. 64
Power of a power: multiply the exponents, 3 × 2 = 6. 2⁶ = 64. Check another way: 2³ = 8 and 8² = 64. The 32 choice comes from adding the exponents (2⁵).
What is the value of 10⁻²?
- 0.01
- −100
- −20
- 0.1
Show answer and explanation
Answer: A. 0.01
A negative exponent means a reciprocal: 10⁻² = 1/10² = 1/100 = 0.01. It does not make the number negative, so −100 and −20 are traps.
Simplify √48.
- 3√4
- 6√2
- 16√3
- 4√3
Show answer and explanation
Answer: D. 4√3
Find the largest perfect square factor of 48: 48 = 16 × 3. So √48 = √16 × √3 = 4√3. Check: 4√3 squared is 16 × 3 = 48. The choice 6√2 squares to 72, not 48.
Write 3.6 × 10⁴ in standard form.
- 3,600
- 360,000
- 36,000
- 0.00036
Show answer and explanation
Answer: C. 36,000
A positive exponent of 4 moves the decimal 4 places to the right: 3.6 → 36,000. The 3,600 choice moves it only 3 places, and 0.00036 moves it the wrong way.
Multiply (6 × 10³)(4 × 10⁵) and write the answer in scientific notation.
- 2.4 × 10⁹
- 2.4 × 10⁸
- 2.4 × 10¹⁵
- 1.0 × 10⁹
Show answer and explanation
Answer: A. 2.4 × 10⁹
Multiply the fronts: 6 × 4 = 24. Add the exponents: 3 + 5 = 8, giving 24 × 10⁸. Rewrite 24 as 2.4 × 10¹, so the result is 2.4 × 10⁹. Stopping at 2.4 × 10⁸ forgets that shift.
Which number is between 7 and 8?
- √45
- √66
- √70
- √55
Show answer and explanation
Answer: D. √55
7² = 49 and 8² = 64, so the number under the root must be between 49 and 64. Only 55 fits. √45 is below 7, and √66 and √70 are above 8.
What is (−2)⁴?
- −16
- 16
- −8
- 8
Show answer and explanation
Answer: B. 16
The parentheses mean the base is −2: (−2)(−2)(−2)(−2). Each pair of negatives gives a positive, so the result is 16. An even power of a negative number is always positive.
Frequently asked questions
Are exponents on Arithmetic Reasoning or Mathematics Knowledge?
Mostly Mathematics Knowledge, which covers "high school mathematics principles" (official ASVAB). AR may use squares inside area problems.
Do I need to know cube roots?
Know the first five cubes (1, 8, 27, 64, 125) so you can recognize simple cube roots like the cube root of 64 = 4.
Why does a negative exponent not give a negative number?
Because the exponent counts division instead of multiplication. 2⁻³ means one divided by 2 three times, which is 1/8, a positive number.
Sources
Checked November 14, 2026. AptiStep is an independent study site; confirm anything that affects your enlistment with a recruiter.