Calculate powers, negative exponents, fractional powers, and large exponents with exact value, expanded notation, and exponent rules.
2 multiplied by itself 8 times equals 256.
Exponentiation is a fundamental mathematical operation involving two numbers: the base (b) and the exponent or power (n), written as bⁿ. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: multiplying n factors of b.
When multiplying powers with the same base, add their exponents: bᵐ · bⁿ = bᵐ⁺ⁿ.
When dividing powers with the same base, subtract exponents: bᵐ / bⁿ = bᵐ⁻ⁿ.
To raise a power to another power, multiply the exponents: (bᵐ)ⁿ = bᵐ·ⁿ.
A fractional exponent represents a radical root: b^(1/n) = ⁿ√b, and b^(m/n) = ⁿ√(bᵐ).
Calculate 2¹⁰ and evaluate 5⁻³ as an exact fraction.
From the quotient rule: bⁿ / bⁿ = bⁿ⁻ⁿ = b⁰. Since any non-zero quantity divided by itself equals 1, b⁰ must equal 1.
In algebra, 0⁰ is considered an indeterminate form. In discrete mathematics, set theory, and computer science, 0⁰ is commonly defined as 1 so that binomial expansions and power series remain consistent.
For whole integers, our engine uses native arbitrary-precision BigInt algorithms to evaluate exact powers beyond standard 64-bit float limits.
Yes! Negative exponents represent reciprocals (1/bⁿ). Decimal exponents represent roots (e.g. 9^0.5 = √9 = 3).
Understanding your calculation
An exponent indicates repeated multiplication for positive integer powers. Negative powers take reciprocals; fractional powers involve roots within the supported number system.
2^5 = 32, and 2^−3 = 1/8 = 0.125.
Zero cannot have a negative power. Negative bases with fractional exponents may require complex numbers, and very large powers can exceed numeric precision.