Laws of Exponents

Pre-Algebra · Axiom Academy

What 2^4 really means — and the handful of rules that let you multiply, divide, and raise powers without ever writing out every factor. In 2^4 , the large number is the base — the factor being repeated — and the small raised number is the exponent — how many times to write it. So 2^4 means four 2's multiplied together. Watch the exponent do its one job: count the factors. 2. Multiplying Powers: Add the Exponents What is ? Write out the factors: . Pushing the two groups together makes one run of five 2's, so with the same base the exponents simply add . 3. Dividing Powers: Subtract the Exponents Division is the reverse. In every 2 on the bottom cancels a 2 on top. Two cancellations leave three 2's behind, so the exponents subtract . 4. A Power of a Power: Multiply the Exponents (2^3)^2 means two copies of 2^3 . Stack them and you get a grid: 2 rows of 3 factors — that is factors in all. 5. A Product Raised to a Power When a whole product is raised to a power, the exponent lands on each factor. is ; regroup the 2's with the 2's and the 3's with the 3's to get . 6. Zero and Negative Exponents: Follow the Pattern Nobody has to declare what 2^0 means — the pattern decides. Count the exponents down and each step divides by 2 : 16, 8, 4, 2, and so on. The step just past 2^1 is 2^0 , and . Keep going below zero and the values become reciprocals . You built every core exponent rule from a single idea — an exponent is a count of equal factors — so none of them has to be memorized cold.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.