Properties of Integer Exponents

Every exponent rule comes from one idea: an exponent counts how many times a base is multiplied by itself. x³ times x² is (x times x times x) times (x times x), which is five factors of x, so you add exponents: x³⁺² = x^5. Dividing powers cancels shared factors, so x⁵ divided by x² leaves three factors of x, matching x^5-2 = x^3. Raising a power to a power, like (x³)⁴, means four groups of three factors, which multiplies the exponents to give x^12.

The zero and negative cases fall out of the same pattern instead of needing new rules. Follow the division pattern down: x³ divided by x³ is x^3-3 = x⁰, but any number divided by itself is 1, so x⁰ = 1. Push one step further: x² divided by x³ = x^2-3 = x⁻¹, and by direct cancellation that's 1/x. A negative exponent never makes the result negative. It flips the base into a fraction, 1/xⁿ, which is small and positive.

This topic sits next to radicals and rational exponents, since x^1/2 is just another case of the same counting logic applied to roots. It also connects to scientific notation, where negative exponents shrink numbers below 1. On the number and quantity items, expect these rules folded into messier expressions with coefficients and several variables, so clean handling of exponents is what keeps simplification fast and accurate.