Exponent rules chart
An exponent says how many times a number is multiplied by itself: 2⁴ is 2 × 2 × 2 × 2, or 16. The exponent rules, also called the laws of exponents, are the shortcuts for multiplying, dividing and raising powers of the same base.
Free printable PDF, 6th grade to 8th grade.
One page, black and white, prints the same every time.
What is on the sheet
The rules
| Rule | In symbols | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ = 625 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)² = 4/9 |
| Zero exponent | a⁰ = 1, when a is not 0 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/2³ = 1/8 |
| Exponent of one | a¹ = a | 9¹ = 9 |
Powers to know
| Power of 2 | Value | Power of 10 | Value |
|---|---|---|---|
| 2¹ | 2 | 10⁰ | 1 |
| 2² | 4 | 10¹ | 10 |
| 2³ | 8 | 10² | 100 |
| 2⁴ | 16 | 10³ | 1,000 |
| 2⁵ | 32 | 10⁴ | 10,000 |
| 2⁶ | 64 | 10⁵ | 100,000 |
| 2⁷ | 128 | 10⁶ | 1,000,000 |
| 2⁸ | 256 | 10⁷ | 10,000,000 |
| 2⁹ | 512 | 10⁸ | 100,000,000 |
| 2¹⁰ | 1,024 | 10⁹ | 1,000,000,000 |
Reading the rules
Every rule on the chart comes from writing the powers out. 2³ × 2⁴ is three 2s times four 2s, which is seven 2s, so the exponents add. 5⁶ ÷ 5² cancels two 5s from the top and the bottom, leaving four, so the exponents subtract. A child who forgets a rule can always get it back this way with small numbers.
The first two rules need the same base. 2³ × 3⁴ cannot be combined into one power, and that is the mistake to watch for on a test.
Which grade uses which rule
Sixth grade writes and evaluates whole number exponents (CCSS 6.EE.A.1), and fifth grade has already met the powers of 10. The rules themselves, including zero and negative exponents, are 8th grade (8.EE.A.1). The powers of 10 table is the one to use for scientific notation, which comes in the same year.
Common questions
- Why is any number to the power of zero equal to 1?
- Follow the quotient rule. 5³ ÷ 5³ is 5⁰, and any number divided by itself is 1, so 5⁰ has to be 1. The rule does not work for 0⁰, which is why the chart says the base is not 0.
- Does a negative exponent make the number negative?
- No. A negative exponent means one over the power: 2⁻³ is 1/8, which is a small positive number. A minus sign in front of the base is what makes a number negative.