Laws of Exponents

Algebra 1 · Axiom Academy

Each rule is just bookkeeping for repeated multiplication — watch the factors get counted, cancelled, and regrouped, and every law falls out on its own. When you multiply two powers of the same base , you're just pushing two groups of factors into one long line. Stack x^ a next to x^ b and the total number of factors is a+b — so the exponents add . Dividing powers of the same base is just cancelling matching factors top-against-bottom. Each factor on the bottom kills one on the top, so leaves a-b factors standing — the exponents subtract . Each bottom factor cancels one on top Raising a power to a power means making b copies of the block x^ a , then expanding every copy into its a factors. You end up with a grid of a columns and b rows, so the total is factors — the exponents multiply . b copies of a factors each → multiply When a whole product is raised to a power, write out the n copies of and then regroup : gather all the x 's together and all the y 's together. Each ends up appearing n times, so the exponent distributes to every factor inside. The exponent lands on each factor These aren't new rules — they're what the quotient rule forces when the bottom count meets or beats the top. Cancel x^ a against itself and everything cancels, leaving 1 ; cancel past zero and the leftover factors land in the denominator. Both come straight from the quotient rule , but a thing over itself is 1 . So x^ 0 =1 for any . , and the three leftover bottom factors give .

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