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

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If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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09 Feb 2016, 09:54
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75% (hard)

Question Stats:

56% (02:07) correct 44% (02:04) wrong based on 223 sessions

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

(1) x < 0
(2) z^z < 1

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If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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09 Feb 2016, 12:10
|x| > 1
|z| > 1
is z^x <1
(1) x < 0
If x = -2 ,
z=2 then z^x = 2^(-2)= 1/(2^2) = 1/4 <1
z=-2 then z^x = (-2)^(-2)=1/(-2^2)=1/4 < 1

if x=-3 ,
z=3 then z^x= 3^(-3) = 1/(3^3) = 1/27 <1
z=-3 then z^x = (-3)^(-3) = 1/(-3^3) = -1/27<1
Sufficient

(2) z^z < 1
=> z is negative , z^x <1 will depend on x .
if z=-2 , then z^z = (-2)^(-2) = 1/(-2^2)= 1/4 <1
if z=-3 , then z^z=(-3)^(-3) = 1/(-3^3) = -1/27<1
However , we have no information about x .
Not sufficient

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Re: If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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09 Feb 2016, 17:04

Statement 1:- X<0 means X is -ve integer. In this case Z^X can be written in the form (1/Z)^X. No no matter what but the result will come in fraction form since Z is an integer and hence the result will be less than 1.

Statement 2:- Z^Z <1 which means either Z is 0 or Z is -ve Odd integer. But it doesn't give any information about X.
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Re: If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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11 Feb 2016, 03:31
1
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

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If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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03 Sep 2017, 00:09
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.

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Re: If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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07 Nov 2017, 12:23

Statement 1:- X<0 means X is -ve integer. In this case Z^X can be written in the form (1/Z)^X. No no matter what but the result will come in fraction form since Z is an integer and hence the result will be less than 1.

Statement 2:- Z^Z <1 which means either Z is 0 or Z is -ve Odd integer. But it doesn't give any information about X.

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
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Re: If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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12 Jan 2019, 01:13
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.

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Re: If z and x are integers with absolute values greater than 1, is z^x  [#permalink]

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12 Jan 2019, 10:22
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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Re: If z and x are integers with absolute values greater than 1, is z^x   [#permalink] 12 Jan 2019, 10:22
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