Exponents & Radicals at the 700-800 Level

Easy exponent questions ask you to simplify. Hard ones hide a conversion step in front of the simplification, then ask for a quantity you were never solving for.

Five archetypes cover the bank. Force a common base — every equation with unequal bases is one base in disguise, and equal bases force equal exponents; converting neither side or only one side both produce clean integers that sit in the choices. Factor out the common power — when powers are added or subtracted you cannot combine exponents, so $3^{x+1} + 3^{x} = 4 \cdot 3^{x}$ turns a hard equation into one line. in modeling formulas. — squaring is not reversible, a principal square root is never negative, and items ask for the of solutions precisely so that skipping the check lands on its own distinct wrong answer. run a four-step machine, each step with a matching distractor.