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

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Math Expert
Joined: 02 Sep 2009
Posts: 52182
If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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27 Apr 2015, 03:28
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85% (hard)

Question Stats:

55% (02:30) correct 45% (02:33) wrong based on 179 sessions

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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.

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Re: If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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27 Apr 2015, 05:08
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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.

1) if $$abc = odd$$ than all three numbers are odd so we can infer that $$z = odd$$
$$abc$$ can be $$+++$$ or $$--+$$ or $$-+-$$ or $$+--$$
Insufficient because we don't know about sign of $$a$$

2) this statement possible than $$b>0$$ and $$c <0$$ or $$b<0$$ and $$c>0$$
But we know nothing about $$z$$ and $$a$$
Insufficient

1+2) we know that $$z$$ is odd
and we know that $$a < 0$$ because
from second statement we know that $$bc = +-$$ or $$bc = -+$$
and from first statement we have two variants: $$abc= --+$$ or $$abc = -+-$$
In both cases $$a <0$$
So $$a^z < 0$$

Sufficient
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Re: If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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28 Apr 2015, 18:23
1
1
a, b, and c are nonzero integers
z = b^c

$$a^z < 0$$ ?
$${a}^{b^c}< 0$$ ?
$${a}^{bc} < 0$$ ?
for the above condition to be true, $$a < 0$$ and $$bc$$ should be odd

Statement (1)
$$abc$$ is an odd positive number;
either $$a$$ and $$bc$$ could be either positive or negative, and $$bc$$ are odd
Not Sufficient

Statement (2)
$$| b + c | < | b | + | c |$$
the above inequality suggests that b and c has opposite signs otherwise $$| b + c | = | b | + | c |$$
so $$bc$$ is negative, but no info about $$a$$
Not Sufficient

from (1) and (2)
$$bc$$ is both negative and odd, so $$a$$ should be negative
hence $${a}^{bc} < 0$$
Sufficient

Math Expert
Joined: 02 Sep 2009
Posts: 52182
If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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04 May 2015, 04:02
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.”

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Re: If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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07 May 2015, 10:22
Hi All,

While this DS question is layered, it's based on a series of Number Properties, so you can solve it with a conceptual approach or by TESTing VALUES (and looking for patterns).

We're told that A, B and C are NON-ZERO INTEGERS and that Z = B^C. We're asked if A^Z is negative. This is a YES/NO question.

Fact 1: (A)(B)(C) is an odd positive number.

This Fact ultimately means that we could have EITHER 0 negative numbers or 2 negative numbers AND that all 3 variables are ODD...

IF.....
A = 1
B = 1
C = 1
Z = 1^1 = 1
A^Z = 1^1 = 1 and the answer to the question is NO.

IF.....
A = -1
B = -1
C = 1
Z = 1^(-1) = 1
A^Z = (-1)^1 = -1 and the answer to the question is YES.

Fact 2: |B + C| < |B| + |C|

This Fact ultimately means that B and C MUST have different signs (one is positive and one is negative)

IF.....
A = 1
B = 1
C = -1
Z = 1^(-1) = 1
A^Z = 1^1 = 1 and the answer to the question is NO.

IF.....
A = -1
B = 1
C = -1
Z = 1^(-1) = 1
A^Z = (-1)^1 = -1 and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we know....
ABC is a positive odd number
|B+C| < |B| + |C|
A, B and C include either 0 negatives or 2 negatives
A, B and C are all ODD
B and C are opposite signs

Since B and C are opposite signs, this means that we MUST have 2 negative numbers (B or C AND A) and 1 positive number (B or C).

A = NEGATIVE ODD

Z = B^C so we either have

B^C = (negative odd)^(positive odd) = NEGATIVE ODD
or
B^C = (positive odd)^(negative odd) = POSITIVE ODD

To boil it down to the essential elements, A^Z = (Negative)^(Odd). The result is ALWAYS NEGATIVE, so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

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Re: If a, b, and c are nonzero integers and z = b^c>, is a^z negative?  [#permalink]

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20 Mar 2018, 05:09
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Re: If a, b, and c are nonzero integers and z = b^c>, is a^z negative? &nbs [#permalink] 20 Mar 2018, 05:09
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