The rule
aⁿ = a multiplied by itself n times
Why it works
A positive whole exponent is repeated multiplication. Extending it to zero, to negatives and to fractions is not arbitrary: each extension is the only one that keeps the rule aᵐ × aⁿ = aᵐ⁺ⁿ working.
How to do it by hand
- For a positive whole exponent, multiply the base by itself that many times
- For a negative one, work out the positive power and take the reciprocal
- For a fractional one, it is a root: a^(1/n) is the nth root
- Anything to the power of zero is 1
What is worth knowing
Why is a⁰ equal to 1? Because aⁿ ÷ aⁿ must be both 1 and a⁰ by the subtraction rule, so the two have to agree. The same argument forces a⁻ⁿ to be 1/aⁿ. Zero to the power of zero is the genuinely contested case: it is conventionally 1 in combinatorics and algebra, and treated as undefined in analysis, and both conventions are defensible. Exponents also grow faster than almost anyone's intuition: 2⁶⁴ grains of rice on a chessboard is more than the world has ever produced.