A student hands in: (i) (2^3)^2 = 2^5; (ii) 5^-1 = -5. The clinic's three questions, in order: What law did they reach for? What does expanded form (or a value) say? What is the correction? (i) They reached for the product law (adding 3 + 2). Expanded form says (2^3)^2 is the 3-factor string repeated twice - 6 factors. Correction: (2^3)^2 = 2^6 = 64. (Their 2^5 = 32 fails the by-values check too: (2^3)^2 = 8^2 = 64.) (ii) They read the exponent's sign as the value's sign. The pattern says a negative exponent is a reciprocal. Correction: 5^-1 = 1/5. (Their -5 fails the size check as well: 5^-1 sits between 5^0 = 1 and 5^-2 = 1/25, so it must be a small positive fraction.) The habit, stated: never argue with a claim in the abstract - re-derive it from expanded form or test it on values.

A student writes (a^2)^4 = a^6. Correct the claim, with the reason -> (a^2)^4 = a^8: the 2-factor string repeats 4 times, so the exponents multiply (2 * 4); a^6 came from adding, the side-by-side law