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

Question

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

Bunuel · Accepted Answer

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

A.  (-a)^{-2b}=\frac{1}{(-a)^{2b}}=positive .

B.  (-a)^{-3b}  --> may be positive (consider a=1 and b=2) as well as negative (consider a=1 and b=1).

C.  -(a^{-2b})=-\frac{1}{a^{2b}}=-\frac{1}{positive}=negative

D.  -(a^{-3b})  --> may be positive (consider a=-1 and b=1) as well as negative (consider a=1 and b=1).

E. None of these

Answer: C.

BrentGMATPrepNow · Answer

IMPORTANT CONCEPT
 Rule #1 : EVEN powers are always greater than or equal to zero. 
So, (POSITIVE value)^(EVEN integer)  >  0, and (NEGATIVE value)^(EVEN integer)  >  0

So, the correct answer here is C. Here's why:

C) – 
Since b is an integer, we know that -2b is an EVEN integer. 
So, we get: – 
By our rule, a^(EVEN integer) is  greater than or equal to zero 
Since a is a NON-zero integer, we can conclude that a^(EVEN integer) is GREATER THAN zero
In other words, a^(EVEN integer) is POSITIVE
This means that -  is NEGATIVE

Answer:  C

Cheers,
Brent

gagan543 · Answer

i have a confusion in option D it is already given a and b is a nonnegative number 
so using same method to solve option D as you used for solving option C 
D would also be always negative  Bunuel

Bunuel · Answer

-(a^{-3b})=-\frac{1}{a^{3b}} .

a^{3b}=a^{odd}  can be negative as well as positive, while  a^{2b}=a^{even}  is always positive.

gagan543 · Answer

i still don't understand, a^3b will always be positive because the question says a & b are not negative so ,so assuming a=1 b=1
a^3(1) will become positive and -(1/positive) will become negative as we cannot take take a=-1 or b=-1

Bunuel · Answer

No, the question says "If a and b are  nonzero  integers, ", NOT non-negative.

ahmedshehabb · Answer

for option C consider a=-2 b=1
-(-2^-2*1)=-(-1/4) the answer would be +ve

Bunuel · Answer

If a = -2, and b = 1, for C we get:

-(a^{-2b})= -((-2)^{-2*1}) = -((-2)^{-2}) = -(\frac{1}{(-2)^2}) = -(\frac{1}{4})=-\frac{1}{4}=negative .

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

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GMAT Problem Solving (PS) Questions

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

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