Exponents and Roots

ACT Math · Axiom Academy

The five rules that govern every power, why zero and negative exponents behave the way they do, and how to simplify radicals — shown as the pattern that makes each rule obvious. 1. Multiplying Powers Is Adding Exponents An exponent just counts how many copies of the base are multiplied together: . Once you see it that way, every exponent rule is really just counting copies — watch two stacks of x 's merge into one taller stack below. Product rule — combine the copies, add the exponents Quotient rule — cancel matching copies, subtract the exponents · · (z^2)^5 = z^ 10 · (2x)^3 = 8x^3 2. Zero and Negative Exponents Are Just the Pattern Continuing Look at powers of 2 as the exponent drops by one each time: . Every step down divides by 2 . Keep dividing — that's the whole rule, not a separate fact to memorize. Watch the ladder descend below. a^0 = 1 (for ) — one more division-by-2 step past 2^1=2 lands exactly on 1 . — the ladder keeps dividing past zero into fractions: . — the denominator names the root, the numerator names the power. — fourth root of 16 is 2, then cubed. 3. Simplifying a Radical Means Splitting Off a Perfect Square A radical asks "what number, raised to the n -th power, gives a ?" To simplify , split a into a perfect square times whatever's left: . Watch split into a clean whole number times a smaller radical. Same idea with a variable: — pull out every perfect-square factor, including even powers of x .

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