Must be Question

[tisrar02]

Could someone help me with this one. Could not find it on the forum. This is from a MGMAT exam. Just need some clarification on the different answer choices as I could not understand the OE.

If a and b are nonzero integers, which of the following must be negative?

A) (-a)^-2b
B) (-a)^-3b
C) -(a^-2b)
D) -(a^-3b)
E) None of these

OA: C

Thanks

[Ian Stewart]

When you raise a nonzero number to an even power, the result is always positive. When you raise a nonzero number to an odd power, its sign doesn't change. It doesn't matter whether the power is positive or negative.

So here, in answer choice A, the exponent is even, so the result must be positive. In answer choice B, the exponent may be odd if b is odd, in which case it won't affect the sign at all. The base is -a, and we don't know if -a is positive or negative; if a is positive, then -a is negative, but if a is negative, then -a is positive. So B is sometimes positive and sometimes negative. In C, we need to evaluate the expression in the brackets first, and since the exponent is even, the expression within the brackets will be positive. But we have a negative sign in front of the brackets, which will make the overall value negative. So C must be negative, and C is the right answer. D is again sometimes positive and sometimes negative; if b is odd, then the exponent is odd and does not affect the sign at all, but we don't know if the base a is positive or negative.

[tisrar02]

When you raise a nonzero number to an even power, the result is always positive. When you raise a nonzero number to an odd power, its sign doesn't change. It doesn't matter whether the power is positive or negative.

So here, in answer choice A, the exponent is even, so the result must be positive. In answer choice B, the exponent is odd, so it won't affect the sign at all. The base is -a, and we don't know if -a is positive or negative; if a is positive, then -a is negative, but if a is negative, then -a is positive. So B is sometimes positive and sometimes negative. In C, we need to evaluate the expression in the brackets first, and since the exponent is even, the expression within the brackets will be positive. But we have a negative sign in front of the brackets, which will make the overall value negative. So C must be negative, and C is the right answer. D is again sometimes positive and sometimes negative; the odd power does not affect the sign at all, but we don't know if the base a is positive or negative.

Thanks for the reply Ian.

So when answering this, I would think this way:
Example A)==> -a^-2b= -1/a^2b... 2b would then make the entire number positive right?

I guess the variables really threw me off.

[GMATGuruNY]

Could someone help me with this one. Could not find it on the forum. This is from a MGMAT exam. Just need some clarification on the different answer choices as I could not understand the OE.

If a and b are nonzero integers, which of the following must be negative?

A) (-a)^-2b
B) (-a)^-3b
C) -(a^-2b)
D) -(a^-3b)
E) None of these

OA: C

Thanks

Try to prove that the answer choices DON'T have to be negative.
To make -a POSITIVE, let a=-1.
To make all of the exponents positive and avoid fractions, let b=-1.
Plugging a=-1 and b=-1 into the answers, we get:

A) (-a)^-2b = 1² = 1.
B) (-a)^-3b = 1³ = 1.
C) -(a^-2b) = -( (-1)² ) = -1.
D) -(a^-3b) = -( (-1)³ ) = 1.
Eliminate A, B and D, since they don't have to be negative.

Now look for a way to make C positive.
-(a^-2b) = -( 1/(a^2b) ).
A nonzero integer (a) raised to an even power (2b) will always yield a POSITIVE result.
Thus, a^2b > 0 and -( 1/(a^2b) ) = -(1/positive) = -(positive) = negative.
Thus, C must be negative.

The correct answer is C.