Exponent Calculator
Calculate any base raised to any power — including negative, fractional, and zero exponents. See the exponent laws and step-by-step explanation.
⬆️ Exponent Calculator
Result
Steps
💡 What are Exponents and How Do They Work?
An exponent (also called a power or index) tells you how many times to multiply a number by itself. In the expression bⁿ, b is the base and n is the exponent. For example, 2⁵ = 2×2×2×2×2 = 32. Exponents are a compact notation for repeated multiplication, just as multiplication is a compact notation for repeated addition.
Key exponent rules: (1) Any number to the power of 0 is 1 (except 0⁰, which is undefined in some contexts). (2) Negative exponents represent reciprocals: b⁻ⁿ = 1/bⁿ. (3) Fractional exponents represent roots: b^(1/n) = ⁿ√b, and b^(m/n) = ⁿ√(bᵐ). (4) Product rule: bᵐ × bⁿ = bᵐ⁺ⁿ. (5) Quotient rule: bᵐ ÷ bⁿ = bᵐ⁻ⁿ. (6) Power of a power: (bᵐ)ⁿ = bᵐⁿ.
Exponents appear everywhere in science and mathematics: compound interest (A = P(1+r)ᵗ), population growth, radioactive decay, pH scale (powers of 10), computing (binary powers of 2), and the decibel scale for sound. Understanding exponent rules is crucial for algebra, calculus, and applied mathematics.
Where:
- b = The base number
- n = The exponent (power)
- b⁰ = 1 = Zero exponent rule (b ≠ 0)
- b⁻ⁿ = 1/bⁿ = Negative exponent = reciprocal
📝 Worked Example
2¹⁰
2×2×2×2×2×2×2×2×2×2= 1,024
5⁻³
1 / 5³ = 1 / 125= 0.008