Exponent Operations
An exponent tells you how many times to multiply a number by itself. This calculator handles powers, roots, exponentials, and logarithms.
Power (xⁿ)
Multiply the base by itself n times.
- 2⁴ = 2 × 2 × 2 × 2 = 16
- 10³ = 10 × 10 × 10 = 1000
- 5⁰ = 1 (any number to power 0 is 1)
- 2⁻³ = 1/2³ = 1/8 = 0.125 (negative exponent = reciprocal)
Root (ⁿ√x)
The inverse of exponentiation. Find what number multiplied n times gives x.
- √16 = 4 (because 4² = 16)
- ∛27 = 3 (because 3³ = 27)
- Fractional exponents are roots: x^(1/n) = ⁿ√x, so 8^(1/3) = ∛8 = 2
Exponential (eˣ)
e ≈ 2.71828 raised to the power x. Appears in compound interest, population growth, radioactive decay, and probability. e¹ = 2.718, e² = 7.389, e⁻¹ = 0.368.
Logarithm
The inverse of exponentiation. log₁₀(x) asks: "10 to what power equals x?"
- log(100) = 2 (because 10² = 100)
- log(1000) = 3 (because 10³ = 1000)
- ln(x) = natural log, using base e instead of 10
Laws of Exponents
These rules simplify expressions involving exponents:
- Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ — multiply same base, add exponents. Example: 2³ × 2⁴ = 2⁷ = 128
- Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ — divide same base, subtract exponents. Example: 5⁶ ÷ 5² = 5⁴ = 625
- Power of a power: (aᵐ)ⁿ = aᵐˣⁿ — multiply exponents. Example: (2³)² = 2⁶ = 64
- Power of a product: (ab)ⁿ = aⁿ × bⁿ — distribute exponent. Example: (3×4)² = 3² × 4² = 144
- Zero exponent: a⁰ = 1 (for any non-zero a)
- Negative exponent: a⁻ⁿ = 1/aⁿ — flip to a fraction
Real-World Applications
- Compound interest: A = P(1 + r/n)^(nt) uses exponents to calculate investment growth
- Scientific notation: 3.0 × 10⁸ m/s (speed of light) — exponents express very large or small numbers
- Computer science: Binary storage uses powers of 2. A 10-bit number stores 2¹⁰ = 1,024 values
- Population growth: Exponential growth P(t) = P₀ × eʳᵗ models bacterial and population growth