Power Calculator

Special cases

• • Negative exponent: a⁻ⁿ = 1 ÷ aⁿ
• • Zero exponent: a⁰ = 1 (for any a ≠ 0)
• • Fractional exponent: a^(1/n) = ⁿ√a
• • Zero base: 0ⁿ = 0 (for n > 0)

Exponentiation: definition and properties

Exponentiation with positive integer exponent is repeated multiplication: aⁿ = a·a·...·a (n times). Example: 2¹⁰ = 1024. Key properties: aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (by convention, with 0⁰ indeterminate), and a⁻ⁿ = 1/aⁿ. Rational exponents connect to roots: a^(p/q) = ᵍ√(a^p). The definition extends to real exponents via the exponential and logarithm: aˣ = exp(x·ln a). Compare growth rates: exponential O(2ⁿ) dominates any polynomial O(nᵏ) for large n — a critical distinction in algorithm analysis (Big-O).

Applications

Compound interest: future value is FV = PV·(1+i)^t. Radioactive decay and population growth follow N(t) = N₀·e^(kt). Computational complexity: algorithms are classified as polynomial (P) or exponential (EXP). Cryptography: RSA and Diffie-Hellman rely on modular exponentiation aᵇ mod n, fast to compute but hard to invert. Scientific notation: physics and chemistry write numbers like 6.022·10²³ (Avogadro) using powers of 10.

FAQ

Why is a⁰ = 1? By the property aᵐ/aᵐ = aᵐ⁻ᵐ = a⁰, and aᵐ/aᵐ = 1 for a ≠ 0. The case 0⁰ is indeterminate (context-dependent).

What does a⁻ⁿ mean? The reciprocal of aⁿ: a⁻ⁿ = 1/aⁿ. Example: 2⁻³ = 1/8 = 0.125.

Can the exponent be irrational? Yes. 2^π is defined via the exponential function: 2^π = exp(π·ln 2) ≈ 8.825.