Power Calculator
Calculate the power aⁿ of any base and exponent, including negatives and decimals. Shows the result in scientific notation when necessary.
Base
Expoente
log₁₀ do resultado
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.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.