Bunuel wrote:

If a, b, and c are nonzero integers and z = b^c>, is a^z negative?

(1) abc is an odd positive number.

(2) | b + c | < | b | + | c |

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:For the expression a^z to be negative, you need the base a to be negative, and you need the exponent to be an odd integer (or a fraction that, when fully reduced, has an odd numerator, such as 1/3 or 5/3, and for that matter an odd denominator; otherwise, you don’t get a real number out).

Since Statement (2) doesn’t contain any reference to a, start with this statement (it’s easier to deal with).

Statement (2): NOT SUFFICIENT. You don’t know whether a is negative or not, so you can be pretty sure at this point that this statement is not sufficient. However, you have to be a little careful – if this constraint on b and c led to a situation in which a^z is never negative, then this statement would be sufficient (answering the question with a no).

This statement is really telling you that b and c have different signs.

If they’re both positive, then | b + c | = | b | + | c |. Likewise, if both b and c are negative, then the two expressions are equal. (Try numbers to see.) The only way for the statement to hold true is for b and c to have different signs.

However, simply knowing that b and c have different signs isn’t enough, as it turns out. If you knew that z (= b^c) would always be an even integer when b and c have different signs, then you’d have sufficiency (because then a^z would never be negative), but that’s not always true.

For instance, if b = -2 and c = 3, then z = -8, but if b = -3 and c = 2, then z = 9.

Statement (1): NOT SUFFICIENT. This statement tells you that a, b, and c are all odd integers, and either all three are positive or exactly one is positive (and the other two are negative). If all three are positive, then the expression a^z is positive. But if a is negative, then you can easily get a negative result: for instance, if a = -1 and z = -27 (from b = -3 and c = 3), then a^z = -1.

Statements (1) and (2) together: SUFFICIENT. First, consider the signs of the variables. You know from (2) that b and c have different signs, so one is positive and one is negative. Since (1) tells you that either all three (a, b, c) are positive or exactly one is positive, then you know that a has to be negative. Moreover, all three are odd. The two possibilities at this point are these:

EITHER a neg odd, b pos odd, c neg odd

OR a neg odd, b neg odd, c pos odd

In the “Either” case, what does z become? A positive odd number raised to a negative odd power becomes a fraction: 1/(odd). For example, \(3^{-5} = (\frac{1}{3})^5 = \frac{1}{243}\). So a (which is negative) raised to that power (of, say, 1/243) will stay negative. (By the way, it’s legal to take an “odd denominator” power of a negative number. You can’t raise a negative number to the power of 1/2, because that’s taking the square root – you don’t get a real-number result – but you can raise a negative number to the power of 1/3, 1/5, or 1/243.)

How about the “Or” case? A negative odd number raised to a positive odd power becomes a negative odd number. For instance, (-3)^5 = -243. So a (which, as before, is definitely negative) raised to that power (of, say, -243) stays negative, since a negative number raised to the power of any odd number (positive or negative) is essentially multiplied by itself an odd number of times, yielding a negative (and then, if the power is negative, you take the reciprocal, which is also negative).

Either way (“Either” or “Or”), the result is negative. The answer to the question is definitely “Yes.”

The correct answer is C.
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