Integers & Number Properties • Topic 3 of 5

Positive/Negative Rules

Signs behave predictably, and the GRE exploits every student who forgets that. For multiplication and division: same signs give a positive result, opposite signs give a negative; an even number of negative factors is positive, an odd number is negative. Powers follow the same logic — a negative base raised to an even power is positive, to an odd power is negative, so (−2)⁴ = 16 but (−2)³ = −8. The subtle traps live in inequalities: multiplying or dividing an inequality by a negative reverses it, and a variable "number" may be negative, zero, or a fraction unless you are told otherwise. On Quantitative Comparison, plugging in a negative or a fraction (not just 2 or 3) is often what exposes a "cannot be determined" answer.

✅ Solved examples

1. Evaluate (−3)² and −3².
(−3)² = (−3)(−3) = 9 (the negative is squared). But −3² means −(3²) = −9 (the power binds before the sign). They differ — read parentheses carefully.
2. What is the sign of (−1)(−2)(−3)(4)?
Count the negative factors: three of them (odd). An odd number of negatives is negative. The value is −24.
3. If −2x > 6, solve for x.
Divide both sides by −2 and FLIP the inequality: x < −3.
4. Quantitative Comparison. x < 0. Quantity A: x³. Quantity B: x². Which is greater?
For any negative x, x³ is negative and x² is positive. A positive is always greater than a negative, so Quantity B is greater.

✏️ Practice — try these, take hints as needed

1. Evaluate (−2)⁴.
Even power of a negative.
Result is positive.
2⁴ = 16.
16
2. What is the sign of the product of five negative numbers?
Count the negatives.
Five is odd.
Odd count → negative.
Negative
3. Solve −5x ≤ 20.
Divide by −5.
Flip the sign.
x ≥ −4.
x ≥ −4
4. If a > 0 and b < 0, what is the sign of a/b?
Opposite signs.
positive / negative.
Result is negative.
Negative
5. Is (−1)^100 − (−1)^101 equal to 0, 1, or 2?
(−1)^even = 1.
(−1)^odd = −1.
1 − (−1).
2

📝 Topic test — 8 questions

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