Exponents and Roots (including Fractional and Negative)

GRE Quantitative · Axiom Academy

One consistent pattern — positive, zero, negative, and fractional exponents are all the same rule, never a special case. 1. One Base, One Unbroken Pattern Watch 2^n as n slides from 4 down to -2 . Each step down halves the value — and that single rule never changes, whether n is a whole number, zero, negative, or a fraction. The bars and the smooth curve y = 2^x are showing the exact same thing. Whole exponent — repeated multiplication Negative exponent — the reciprocal Reading the pattern in numbers Example: 5^1 = 5 , and 1^n = 1 for any n 2. A Root Is the Power, Undone A square root doesn't need its own separate rulebook — it's just the inverse of squaring. Slide a point along y = x^2 and watch its mirror image, reflected across the line y = x , trace out at exactly the same instant. That's what a fractional exponent is . — the denominator n is the root. — take the root, then raise it. only — never . The principal root is always non-negative. , since odd roots exist for negative numbers too. The GRE loves asking "which is bigger?" Sweep x from 0 to 2 across y = x , y = x^2 , and y = x^3 : below 1 , squaring and cubing shrink the number, so x > x^2 > x^3 . Above 1 , the same operations grow it, flipping the order to x < x^2 < x^3 . The crossover happens exactly at x = 1 , where all three meet. x > x^2 > x^3 — repeated multiplication by a fraction makes small numbers smaller. x < x^2 < x^3 — repeated multiplication by a number over 1 makes it grow.

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