A student evaluates 3⁻² and writes −9. What error did they make, and what is the correct answer?
AThey forgot to square first; the correct answer is −6
BThey confused a negative exponent with a negative base; the correct answer is 1/9
CThey applied the wrong base; the correct answer is 1/6
DThey reversed the sign; the correct answer is 9
3⁻² = 1/3² = 1/9. The student treated the negative exponent as a negative sign on the result, computing −(3²) = −9 instead. This is the core misconception: a negative exponent signals a reciprocal, not a sign change. The base's sign determines the sign of the result; the exponent's sign determines whether the result is the base-power or its reciprocal.
Question 2 Multiple Choice
Which expression is equivalent to (2x⁻³) / (y⁻²)?
A−2x³ / y²
B2 / (x³y²)
C2y² / x³
D−2y² / x³
Apply the 'move and flip' rule to each negative exponent: x⁻³ in the numerator moves to the denominator as x³; y⁻² in the denominator moves to the numerator as y². The 2 is unaffected (no negative exponent). Result: 2y² / x³. The negative exponents never make the result negative — they signal reciprocals. Options A and D introduce negative signs that have no basis in the rules.
Question 3 True / False
A negative exponent makes the result a negative number.
TTrue
FFalse
Answer: False
This is the central misconception. x⁻² = 1/x², which is positive for any nonzero x. 2⁻³ = 1/8, not −8. A negative exponent signals a reciprocal operation, not a sign change. The sign of the result is determined entirely by the sign of the base (and whether the exponent is even or odd), not by whether the exponent is negative. 'Negative exponent' and 'negative result' are unrelated concepts.
Question 4 True / False
The rule x⁻ⁿ = 1/xⁿ follows necessarily from requiring the quotient rule (x^a / x^b = x^(a−b)) to remain consistent when a is less than b.
TTrue
FFalse
Answer: True
The derivation: x²/x⁵ computed by cancellation gives 1/x³. The same expression computed by the quotient rule gives x^(2−5) = x^(−3). For both results to be equal, x^(−3) must equal 1/x³. This is not an arbitrary definition — it is the only definition that keeps the quotient rule consistent for all integer exponents. The rule is forced by the logic of the existing exponent rules.
Question 5 Short Answer
Explain how the sequence x³, x², x¹, x⁰, x⁻¹, x⁻², ... (each step dividing by the base) makes negative exponents feel natural rather than arbitrary.
Think about your answer, then reveal below.
Model answer: Each step to the left in the sequence divides by the base. Starting from x² = x·x, dividing by x gives x¹ = x. Dividing again gives x⁰ = 1. Dividing again gives x⁻¹ = 1/x. Dividing again gives x⁻² = 1/x². The pattern extends naturally — negative exponents are simply what you get when you keep dividing past zero. They are not a new concept but the continuation of the same pattern that produced x⁰ = 1.
The sequence visualization makes negative exponents feel inevitable rather than invented. Students who only see the abstract rule x⁻ⁿ = 1/xⁿ often treat it as memorizable fact; students who see it as the continuation of the dividing-by-x pattern understand why the rule must be what it is. This also connects x⁰ = 1 (which often feels arbitrary) to the same underlying logic.