Exponent Calculator Guide: Exponent Rules Explained (2026)
Quick Answer
- *Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- *Zero exponent: a° = 1 (for a ≠ 0)
- *Negative exponent: a⁻ⁿ = 1/aⁿ (reciprocal, not negative)
- *Fractional exponent: a^(m/n) = the nth root of aᵐ
The Seven Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ = 32 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power | (aᵐ)ⁿ = aᵐ×ⁿ | (2³)² = 2⁶ = 64 |
| Zero | a° = 1 | 7° = 1 |
| Negative | a⁻ⁿ = 1/aⁿ | 5⁻² = 1/25 |
| Product of Powers | (ab)ⁿ = aⁿbⁿ | (2×3)² = 4×9 = 36 |
| Quotient of Powers | (a/b)ⁿ = aⁿ/bⁿ | (4/3)² = 16/9 |
Why a° = 1
Consider the pattern: 2³ = 8, 2² = 4, 2¹ = 2. Each step divides by 2. The next step: 2° = 2 ÷ 2 = 1. Formally, the quotient rule proves it: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a°, and anything divided by itself is 1.
Negative Exponents
A negative exponent means “take the reciprocal.” It does not make the result negative.
- 2⁻³ = 1/2³ = 1/8 = 0.125
- 10⁻² = 1/100 = 0.01
- (−3)⁻² = 1/(−3)² = 1/9
Fractional Exponents
The denominator of a fractional exponent is the root; the numerator is the power: a^(m/n) = ⁿ√(aᵐ).
- 27^(1/3) = ³√27 = 3
- 16^(3/4) = (&sup4;√16)³ = 2³ = 8
- 8^(2/3) = (³√8)² = 2² = 4
Scientific Notation
Scientific notation uses powers of 10 to express very large or small numbers compactly:
- Speed of light: 300,000,000 m/s = 3 × 10&sup8; m/s
- Hydrogen atom radius: 0.000000000053 m = 5.3 × 10⁻¹¹ m
- US national debt: ~$36,000,000,000,000 = 3.6 × 10¹³
Common Mistakes
- Confusing (−3)² and −3²: (−3)² = 9 but −3² = −9. Parentheses matter.
- Adding exponents when multiplying different bases: 2³ × 3² ≠ 6⁵. The product rule only works with the same base.
- Thinking negative exponents give negative results: 2⁻³ = 1/8, not −8.
Compute any exponent expression
Try the Free Exponent Calculator →Frequently Asked Questions
What are the basic rules of exponents?
Seven rules: product (add exponents), quotient (subtract exponents), power (multiply exponents), zero (result is 1), negative (take reciprocal), product of powers, and quotient of powers.
Why is anything to the power of 0 equal to 1?
From the quotient rule: aⁿ ÷ aⁿ = a°, and any non-zero number divided by itself equals 1. The pattern also shows it: 2³=8, 2²=4, 2¹=2, 2°=1.
How do negative exponents work?
They mean “take the reciprocal”: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/8. The result is a fraction, not a negative number.
What is scientific notation?
A number between 1 and 10 multiplied by a power of 10. 6,380,000 = 6.38 × 10⁶; 0.00042 = 4.2 × 10⁻&sup4;.
How do fractional exponents work?
The denominator is the root, the numerator is the power: a^(m/n) = ⁿ√(aᵐ). Example: 8^(2/3) = (³√8)² = 4.