Exponent Rules & Properties

Calculus Readiness · Axiom Academy

Every law of exponents is just bookkeeping for repeated multiplication — watch the factors do the counting for you. If you ever forget a rule, expand the powers as factors and it reappears. 1. Product Rule — multiply, and the exponents add Multiply two powers of the same base and you are just piling their factors into one row. Count them: three a 's next to two a 's is five a 's. So the exponents add . 2. Quotient Rule — divide, and the exponents subtract Dividing powers of the same base lets each factor on the bottom cancel one on top. Whatever survives is the difference. So the exponents subtract . The numerator a^m contributes m copies of the base. The denominator a^n contributes n copies — each one cancels a copy above. After cancelling, m - n factors remain on top: that is a^ m-n . If n > m , leftover factors sit underneath — a negative exponent (Step 5). 3. Power Rule — a power of a power multiplies Raising a^m to the n means writing that block of m factors down n times. Stack them into a grid: m across, n down. The total count of factors is — so the exponents multiply . Why it isn't addition: , and the product rule then adds 3 + 3 = 6 . Repeated addition of m , done n times, is exactly . (x^4)^3 = x^ 12 · (2^2)^5 = 2^ 10 = 1024 · (y^ -3 )^2 = y^ -6 4. Zero Exponent — why a^0 = 1 Step down the powers of a base and each step divides by that base . Walk — halving every time. One more halving lands on 2^0 = 1 . The pattern forces the value of the zero power.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.