An exponent is shorthand for repeated multiplication: in bⁿ, the base b is multiplied by itself n times. The idea extends smoothly beyond whole numbers — a zero exponent gives 1, a negative exponent gives the reciprocal (b⁻ⁿ = 1/bⁿ), and a fractional exponent gives a root (b^(1/n) = ⁿ√b). These extensions exist precisely so the laws of exponents — add when multiplying, subtract when dividing, multiply when nesting — keep working everywhere.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the laws of exponents.
The exponent rules
The three working laws follow directly: bᵐ × bⁿ = b^(m+n), bᵐ ÷ bⁿ = b^(m−n), and (bᵐ)ⁿ = b^(m·n). The division law explains b⁰ = 1 (since bⁿ ÷ bⁿ = b⁰) and the negative exponent (since b⁰ ÷ bⁿ = b⁻ⁿ = 1/bⁿ). Fractional exponents fill in the gaps between integers: because (b^(1/n))ⁿ = b¹ = b, the quantity b^(1/n) must be the n-th root of b. That is why √b and b^0.5 are the same thing.
Worked example — powers of two and more
Positive: 2¹⁰.
Negative & fractional: 2⁻³ and 9^½.
So 2¹⁰ = 1024 — ten doublings, the reason 2¹⁰ is the basis of "kilo" in computing (1024 bytes). The negative power 2⁻³ flips to 1/8 = 0.125, and the half power of 9 is its square root, 3. The same machine handles other everyday powers: 3⁴ = 81, 10⁶ = 1,000,000, and 5³ = 125. Whenever the result runs very large or very small, the calculator also expresses it in scientific notation so it stays readable.
Frequently Asked Questions
How many times to multiply the base by itself. 2¹⁰ = 1024; 5³ = 125.
A reciprocal: b⁻ⁿ = 1/bⁿ. So 2⁻³ = 1/8 = 0.125 — not a negative number.
A root: b^(1/n) = ⁿ√b. 9^½ = √9 = 3; 27^⅓ = 3.
Because bⁿ ÷ bⁿ = b⁰ and anything ÷ itself is 1. (0⁰ is left undefined.)
bᵐ·bⁿ = b^(m+n); bᵐ/bⁿ = b^(m−n); (bᵐ)ⁿ = b^(mn).