If z and x are integers with absolute values greater than 1, is z^x

for integers with absolute value >1 for z^x to be less than 1
2 possibilities:
- x is positive and odd, and z is negative
- or x is negative

Statement 1 says that x is negative = sufficient
Statement 2 says that z is negative = not sufficient as if x positive and even the solution will be greater than 1

Answer A

z and x are integers with absolute values greater than 1, that means z and x can not take 1,0,-1 as values.
we need to check whether z^x is less than 1

stmt 1) x < 0
means z can take any value except 1,0,-1 and x can take negative values.
case 1) z is positive
lets z= 2 and x= -3
(2)^(-3) = 1/(2^3) which is less than 1.
case 2) z is negative
lets z=-2 and x= -3
(-2)^(-3) = 1/ (-2)^(-3) = -1/8 which is less than 1.
Hence statement 1 is sufficient.

stmt 2) z^z < 1
means z must be negative.
x can take any value except 1,0,-1
case 1) z is negative and x is positive
lets z= -2 and x= 4
(-2)^4= 16 which is not less than 1
case 2) z is negative and x is negative
lets z= -2 and x=-4
(-2)^(-4) = 1/16 which is less than 1.
from statement 2, we can not firmly state whether z^x is less than 1.
Hence Statement 2 is not sufficient.

Hence Answer is A

Kudos if it helps

Hi Divya,
Statement 2: z need not be -ve odd integer, z can be -2 also, in which case (-2)^(-2) is still less than 1. But this statement is insuff

Let's analyse the question statement
z & x are integers. And |z| & |x|>1. This means z & x are not equal to 0. Thus, z & x are positive or negative integers and not equal to 0.

Now, \(z^x\) <1 only in following cases:
a) z>0 and x<0: i.e. z=2 & x=-2. Which means \(z^x\)=\(2^-2\)=\(1/4\)<1
b) z<0 and x<0: i.e. z=-2 & x=-2. Which means \(z^x\)=\(-2^-2\)=\(1/4\)<1 or z=-2 & x=-1. Which means \(z^x\)=\(-2^-1\)=\(-1/2\)<1
c) z<0 and x is positive odd integer. Which means \(z^x\)=\(-2^1\)=\(-2\)<1

With these in mind, let's check statements.

1) x<0, this means that first two cases discussed above are applicable. Checking on those, we can understand that for any value of z (positive or negative), \(z^x\)<1, given x<0. This statement is sufficient

2) \(z^z\) < 1. Applying the three cases discussed above, we can sum up that either this condition holds true only if z is negative. But, for \(z^x\)<1, we need to know value of x. Thus, this statement is insufficient.

Hence, answer is A

Answer is A. Just by looking at the question you realize that since x and z are integers, the only way in which you can answer yes is if X is negative, in order to give you the reciprocal.

It doesn't matter if Z is positive or negative.

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