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. Rational-exponent scaling in modeling formulas. Radical equations manufacture fake solutions — squaring is not reversible, a principal square root is never negative, and items ask for the sum of solutions precisely so that skipping the check lands on its own distinct wrong answer. Fractional and negative exponents run a four-step machine, each step with a matching distractor.

The governing failure is stopping early or stopping late. The most common wrong answer in this topic is a number that is genuinely correct — just not the one asked for. Before computing, write the requested expression at the top of your scratch space, then make your last line match that symbol.