jeeteshsingh wrote:
Bunuel wrote:
If a and b are integers, and a not= b, is |a|b > 0?
(1) |a^b| > 0
(2) |a|^b is a non-zero integer.
\(|a|b > 0\) is true when \(b>0\) and \(a\) does not equal to zero.
(1) \(|a^b| > 0\) --> \(a\) does not equal to zero, but we don't know about \(b\), it can be any value, positive or negative. Not sufficient.
(2) \(|a|^b\) is a non-zero integer --> \(a\) can be 1 and \(b\) any integer, positive or negative. Not sufficient.
(1)+(2) If a=1 and b=2, then |a|b > 0, but if a=1 and b=-2, then |a|b <0. Not sufficient.
Answer: E.
This is the easiest way to solve this problem. In my previous solution I just tried to show what each statement means algebraically. Please tell which part needs more clarification.
Not able to get why do you say this (marked in red)... I know that \(|a|\) would always give a positive value, but why cant we consider \(a\) as 2 or 3...? Any reason for this?
OK. For statement (2): First of all I'm not saying that \(a\) can ONLY be 1. I'm saying that it MAY be 1 and in this case \(b\) can take any value statement (2): "
\(|a|^b\) is a non-zero integer " to hold true. So \(b\) can take positive as well as negative values, hence \(|a|b > 0\) may or may not be true. That's why (2) is not sufficient.
To elaborate further: we are told that \(|a|^b\) is
A. not zero and
B. it's an integer.
A. Means that \(a\) is not zero, as
the only way \(|a|^b\) to be zero, is \(a\) to be zero.
B. \(|a|^b\) is an integer (non=zero). What does that mean?
When \(a\) is any integer so that \(|a|>1\) (eg 2, -2, 3, -3, ...), \(b\) can take any value but negative. Example: \(a=-3\) --> \(|-3|=3\), \(3\) in integer power (given \(a\) and \(b\) are integers) to be an integer, power (\(b\)), must be positive or zero, as when \(b\) is negative, let's say \(-2\), we'll have: \(3^{-2}=\frac{1}{3^2}\) which is not integer.
But when \(|a|=1\) (\(1\) or \(-1\)), then \(b\) can take ANY integer value as 1 in any power equals to 1 which is integer.
So from statement (2) we'll have:\(a\) does not equals to 0. Plus, \(b\) can be ANY integer (positive or negative or zero) when \(|a|=1\) and \(b\) must be positive or zero when \(|a|>1\).
Hope it's clear.