Exponent Calculator
Calculate base raised to exponent power xʸ.
2¹⁰ = 1024
The rules all follow from repeated multiplication
An exponent counts repeated multiplication: 2⁵ is five 2s multiplied together. Every rule that follows is a consequence of that, not a separate fact to memorise.aᵐ × aⁿ = aᵐ⁺ⁿ because you are lining up m copies next to n copies. Dividing subtracts for the same reason, and (aᵐ)ⁿ = aᵐⁿ because you take m copies n times over.
The strange-looking cases come from continuing the pattern downward.a⁰ = 1 because aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰, and anything divided by itself is 1. Negative exponents are reciprocals — 2⁻³ = 1/8 — because each step down divides by the base again. Fractional exponents are roots:a^(1/2) is the square root, since squaring it must return a.
Two traps. −2⁴ and (−2)⁴ are different: the first negates2⁴ to give −16, the second raises −2 to give +16, because exponentiation binds tighter than the minus sign. And (a + b)ⁿ is emphatically notaⁿ + bⁿ — exponents distribute over multiplication, never over addition.
Frequently Asked Questions
How do negative exponents work?
A negative exponent x^(-y) is equal to 1 / (x^y).
What is any number raised to the 0 power?
Any non-zero number raised to power 0 equals 1.
What are the rules of exponents?
Multiplying adds exponents (a^m x a^n = a^(m+n)); dividing subtracts them; a power of a power multiplies them; a^0 = 1 for any non-zero a; and a negative exponent is a reciprocal, so a^-n = 1/a^n. A fractional exponent is a root: a^(1/2) is the square root.
Why is anything to the power of zero equal to one?
Because of the division rule: a^n / a^n is obviously 1, and by subtracting exponents it is also a^0. Both must be true, so a^0 = 1. The expression 0^0 is a special case, usually defined as 1 in combinatorics and left undefined in analysis.