Exponents: Powers, Roots, and the Laws That Govern Them
An exponent (or power) tells you how many times to multiply a base by itself. Exponents appear in compound interest, population growth, physics, chemistry, and computing — any time quantities scale multiplicatively.
Laws of Exponents
Product: aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power: (aᵐ)ⁿ = aᵐⁿ
Zero: a⁰ = 1 (a ≠ 0)
Negative: a⁻ⁿ = 1/aⁿ
Fraction: a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)ᵐScientific Notation
Standard form: coefficient × 10ⁿ (coefficient: 1 ≤ x < 10)
93,000,000 = 9.3 × 10⁷
0.0000045 = 4.5 × 10⁻⁶
Multiply: (3×10⁴) × (2×10³) = 6×10⁷Negative and Fractional Exponents
- 2⁻³ = 1/2³ = 1/8 = 0.125
- 8^(1/3) = ∛8 = 2
- 27^(2/3) = (∛27)² = 3² = 9
- 4^(−1/2) = 1/√4 = 0.5
Calculate powers and exponents: Free Exponent Calculator
Exponent Rules
- Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (e.g., 2³ × 2⁴ = 2⁷ = 128)
- Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ
- Power rule: (aᵐ)ⁿ = aᵐⁿ
- Zero exponent: a⁰ = 1 (for any a ≠ 0)
- Negative exponent: a⁻ⁿ = 1/aⁿ (e.g., 2⁻³ = 1/8)
- Fractional exponent: a^(1/n) = ⁿ√a (e.g., 8^(1/3) = ∛8 = 2)
Scientific Notation and Powers of 10
Scientific notation expresses very large or small numbers as a coefficient (1 to 10) times a power of 10. The speed of light: 3 × 10⁸ m/s. An electron mass: 9.109 × 10⁻³¹ kg. Avogadro's number: 6.022 × 10²³ molecules/mol. This notation is essential in physics, chemistry, and astronomy where numbers span many orders of magnitude. Multiplying numbers in scientific notation: (3 × 10⁸) × (2 × 10⁴) = 6 × 10¹² — add the exponents and multiply the coefficients.
Frequently Asked Questions
What is 0⁰?
This is a mathematical controversy. In combinatorics and most applied contexts, 0⁰ = 1 by convention (it makes the binomial theorem and power series work correctly). In analysis, the limit of xˣ as x→0 equals 1. Some mathematicians consider it an indeterminate form. For most calculator and computing purposes, 0⁰ = 1.
How does compound interest use exponents?
The compound interest formula A = P(1 + r/n)^(nt) uses an exponent. P = principal, r = annual rate, n = compounding periods per year, t = years. £1,000 at 5% compounded annually for 10 years: A = 1000 × 1.05¹⁰ = £1,628.89. The exponent captures exponential growth — the growth rate itself grows over time.
What is the difference between 2³ and 3²?
2³ = 2 × 2 × 2 = 8 (two cubed). 3² = 3 × 3 = 9 (three squared). Exponentiation is not commutative: aᵇ ≠ bᵃ in general. The base and exponent play very different roles.