What an exponent does
An exponent is repeated multiplication of the base by itself:
bⁿ = b × b × … × b (n times)
So 2¹⁰ = 2 × 2 × … (ten 2s) = 1,024. The base is the number; the exponent is how many times it appears.
Negative and fractional exponents
- Negative: b⁻ⁿ = 1 ÷ bⁿ. So 2⁻³ = 1 ÷ 8 = 0.125.
- Fraction: a root. 9^(1/2) = √9 = 3, and 8^(1/3) = ∛8 = 2.
- Zero: any non-zero base to the 0 power equals 1.
The laws of exponents
- Product: bᵐ × bⁿ = bᵐ⁺ⁿ
- Quotient: bᵐ ÷ bⁿ = bᵐ⁻ⁿ
- Power of a power: (bᵐ)ⁿ = bᵐⁿ
Frequently asked questions
How do exponents work?
An exponent tells you how many times to multiply the base by itself. So 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is the number being multiplied, and the exponent (or power) is how many times.
What does a negative exponent mean?
A negative exponent means the reciprocal of the positive power: b⁻ⁿ = 1 ÷ bⁿ. For example, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125. The bigger the negative exponent, the smaller the result.
What does a fractional exponent mean?
A fractional exponent is a root. b^(1/2) is the square root of b, and b^(1/3) is the cube root. More generally, b^(m/n) is the n-th root of b raised to the m-th power. So 8^(1/3) = 2 and 9^(0.5) = 3.
What is anything to the power of zero?
Any non-zero number raised to the power of 0 equals 1. This keeps the rules of exponents consistent — for instance, dividing bⁿ by bⁿ gives b⁰, which must equal 1. By common convention, 0⁰ is also treated as 1.
Note: Very large or very small results are shown in scientific notation, and extremely large powers may lose precision in the final digits.