Exponent Rules & Powers
REFExponent Laws
| Rule Name | Rule | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ = 32 |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power Rule | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of Product | (ab)ⁿ = aⁿbⁿ | (2×3)² = 2²×3² = 36 |
| Power of Quotient | (a/b)ⁿ = aⁿ/bⁿ | (4/2)³ = 4³/2³ = 8 |
| Zero Exponent | a⁰ = 1 | 5⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/2³ = 1/8 |
| Fractional Exponent | a¹/ⁿ = ⁿ√a | 16¹/² = √16 = 4 |
Powers of 2
| Power | Value | Power | Value | Power | Value |
|---|---|---|---|---|---|
| 2⁰ | 1 | 2⁵ | 32 | 2¹⁰ | 1,024 |
| 2¹ | 2 | 2⁶ | 64 | 2¹¹ | 2,048 |
| 2² | 4 | 2⁷ | 128 | 2¹² | 4,096 |
| 2³ | 8 | 2⁸ | 256 | 2¹⁵ | 32,768 |
| 2⁴ | 16 | 2⁹ | 512 | 2²⁰ | 1,048,576 |
Powers of 10
| Power | Value | Name |
|---|---|---|
| 10⁰ | 1 | One |
| 10¹ | 10 | Ten |
| 10² | 100 | Hundred |
| 10³ | 1,000 | Thousand |
| 10⁶ | 1,000,000 | Million |
| 10⁹ | 1,000,000,000 | Billion |
| 10¹² | 1,000,000,000,000 | Trillion |
| 10⁻¹ | 0.1 | Tenth |
| 10⁻² | 0.01 | Hundredth |
| 10⁻³ | 0.001 | Thousandth |
Common Powers Table
| Base | ² | ³ | ⁴ | ⁵ |
|---|---|---|---|---|
| 2 | 4 | 8 | 16 | 32 |
| 3 | 9 | 27 | 81 | 243 |
| 4 | 16 | 64 | 256 | 1,024 |
| 5 | 25 | 125 | 625 | 3,125 |
| 6 | 36 | 216 | 1,296 | 7,776 |
| 7 | 49 | 343 | 2,401 | 16,807 |
| 8 | 64 | 512 | 4,096 | 32,768 |
| 9 | 81 | 729 | 6,561 | 59,049 |
| 10 | 100 | 1,000 | 10,000 | 100,000 |
Understanding Exponents
An exponent indicates how many times a base number is multiplied by itself. For example, 2³ = 2×2×2 = 8. Exponents follow specific laws that simplify calculations: when multiplying same bases, add exponents; when dividing, subtract exponents. Zero exponent always equals 1. Negative exponents represent reciprocals. Exponents are fundamental in algebra, science, and computer science (especially powers of 2 in computing).
Frequently Asked Questions
Why does any number to the power of zero equal 1?
Using the quotient rule, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. Since any nonzero number divided by itself is 1, a⁰ must equal 1. For example, 5⁰ = 1.
What does a negative exponent mean?
A negative exponent is the reciprocal of the positive power: a⁻ⁿ = 1/aⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125. It does not make the result negative.
Why are powers of 2 important in computing?
Computers store data in binary, so capacities follow powers of 2: 2¹⁰ = 1,024 (a kibibyte), 2²⁰ = 1,048,576 (a mebibyte). That is why memory sizes appear as 256, 512, 1,024, and so on.