How Exponents Work
An exponent counts repeated multiplication: xⁿ multiplies the base x by itself n times. Exponents follow a small set of rules that make them easy to combine — and those same rules extend naturally to zero, negative and fractional powers.
x⁰ = 1 · x⁻ⁿ = 1/xⁿ · x^(1/n) = ⁿ√x
Compute 3⁴ × 3²: add the exponents — 3⁴⁺² = 3⁶ = 729. Doing it the long way (81 × 9) confirms it.
Powers of 2 and 10
| n | 2ⁿ | 10ⁿ |
|---|---|---|
| 1 | 2 | 10 |
| 2 | 4 | 100 |
| 3 | 8 | 1,000 |
| 5 | 32 | 100,000 |
| 8 | 256 | 100,000,000 |
| 10 | 1,024 | 10,000,000,000 |
Powers of 2 underpin computing (kilobytes, IP addresses, chess-board legends); powers of 10 underpin scientific notation, where 6.02 × 10²³ is far easier to write than 23 digits.
Growth Intuition
Exponential growth outruns intuition quickly. Folding a 0.1 mm sheet of paper 42 times — if you could — would make it 2⁴² × 0.1 mm thick: about 440,000 km, past the Moon. That's why exponents matter for understanding compound interest, population growth and viral spread.
Frequently Asked Questions
What does an exponent mean?
An exponent tells you how many times to multiply the base by itself: 2⁵ = 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 the reciprocal of the positive power: x⁻ⁿ = 1 ÷ xⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125.
What is anything to the power of zero?
Any non-zero number raised to the power 0 equals 1. This keeps the exponent rules consistent: x³ ÷ x³ = x³⁻³ = x⁰, and any number divided by itself is 1. The expression 0⁰ is left undefined in most contexts.
What do fractional exponents mean?
A fractional exponent is a root: x^(1/2) is the square root of x, and x^(1/3) is the cube root. More generally, x^(m/n) is the n-th root of x raised to the m-th power — 8^(2/3) = (∛8)² = 2² = 4.
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The CalculatorsGuide Editorial Team. (2026). Exponent Calculator. CalculatorsGuide. Retrieved from https://www.calculatorsguide.com/math/exponent-calculator/