Exponent rules
| Rule | Example |
|---|
| xᵃ × xᵇ = xᵃ⁺ᵇ | 2³ × 2² = 2⁵ = 32 |
| xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 5⁶ ÷ 5⁴ = 5² = 25 |
| (xᵃ)ᵇ = xᵃᵇ | (3²)³ = 3⁶ = 729 |
| x⁰ = 1 | 7⁰ = 1 |
| x⁻ⁿ = 1 ÷ xⁿ | 10⁻² = 0.01 |
| x^(1/n) = ⁿ√x | 27^(1/3) = 3 |
Solving for the base or exponent
To find the base, take the n-th root: if x⁴ = 81, then x = 81^(1/4) = 3. To find the exponent, use logarithms: if 2ⁿ = 1024, then n = log(1024) ÷ log(2) = 10.
Exponents in real life
Last reviewed October 2026.
Frequently asked questions
How do you calculate an exponent?
An exponent tells you how many times to multiply the base by itself. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent (or power) is 5.
What does a negative exponent mean?
A negative exponent means "one divided by" the positive power. 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125.
What is a fractional exponent?
A fractional exponent is a root. x^(1/2) is the square root of x, x^(1/3) is the cube root, and x^(2/3) is the cube root of x, squared. So 8^(2/3) = 4.
Why is anything to the power of 0 equal to 1?
Each time you lower the exponent by one, you divide by the base: 2³ = 8, 2² = 4, 2¹ = 2, so 2⁰ = 1. The pattern works for any nonzero base. 0⁰ is a special case usually defined as 1 in algebra.
Why can't I raise a negative number to some fractional powers?
A negative base with a fractional exponent like (−8)^(1/2) asks for the square root of a negative number, which is not a real number. Odd roots like (−8)^(1/3) = −2 do exist, and the calculator handles them.