Working with powers
Interactive reference of exponent rules (product, quotient, power of a power) with examples using your numbers.
Why the rules hold
The rules for working with powers aren't arbitrary – they follow directly from the definition of a power as repeated multiplication. Writing out xⁿ · xᵖ, we see (x·x·…·x) · (x·x·…·x), n + p factors in total, i.e. xⁿ⁺ᵖ. The other rules can be derived similarly.
What they are used for
Knowing the rules greatly simplifies calculations – instead of raising large numbers to powers, you just add or multiply exponents. They're used in simplifying algebraic expressions, solving equations, physics formulas, and anywhere exponents and scientific notation are used.
Tip: the rules also hold for negative and decimal exponents, and for roots (a root = a power with exponent 1/n).