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0^0 is equal to 1 in most mathematical conventions, but not all. Therefore, you can expect that GMAT not to test the subject--it won't show up, and you won't be expected to know it!
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QUESTION YOU ARE TALKING ABOUT SHOULD READ: If \(J\neq{0}\), what is the value of \(J\) ?

(1) \(|J| = J^{-1}\) (2) \(J^J = 1\)

Two reasons why should the stem state that \(J\neq{0}\): For statement (1) if \(J=0\) then we'll have \(0^{-1}=\frac{1}{0}=undefined\). Remember you can't raise zero to a negative power. For statement (2) if \(J=0\) then we'll have \(0^0\). 0^0, in some sources equals to 1, some mathematicians say it's undefined. Anyway you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT. So on the GMAT the possibility of 0^0 is always ruled out.

Also notice that saying in the stem that J is an integer is a redundant.

AS FOR THE SOLUTION: If \(J\neq{0}\), what is the value of \(J\) ?

(1) \(|J| = J^{-1}\) --> \(|J|*J=1\) --> \(J=1\) (here J can no way be a negative number, since in this case we would have \(|J|*J=positive*negative=negative\neq{1}\)). Sufficient.

(2) \(J^J = 1\) --> again only one solution: \(J=1\). Sufficient.

I have a doubt in this solution (could be dumb ) but im still asking. I landed up with Option B , but how is Statement 1 alone sufficient ? I mean can't fractions also be considered in statement 1 and couldnt Abs value of 1/2 = 2^-1 ? (i mean isnt that mathematical right) All i could say was that J is definitely positive .. I got 1 as well but I had this fraction option too so didn't go for Statement 1. Is it wrong to consider the fraction ? i dont know if im trying to be too analytical over this but don't want to make this mistake going into the exam Could some one pls pls help.

I have a doubt in this solution (could be dumb ) but im still asking. I landed up with Option B , but how is Statement 1 alone sufficient ? I mean can't fractions also be considered in statement 1 and couldnt Abs value of 1/2 = 2^-1 ? (i mean isnt that mathematical right) All i could say was that J is definitely positive .. I got 1 as well but I had this fraction option too so didn't go for Statement 1. Is it wrong to consider the fraction ? i dont know if im trying to be too analytical over this but don't want to make this mistake going into the exam Could some one pls pls help.

No, the math is not correct. If J = 1/2, then: \(|J| = \frac{1}{2}\) but \(J^{-1}=(\frac{1}{2})^{-1}=2\).
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I have a doubt in this solution (could be dumb ) but im still asking. I landed up with Option B , but how is Statement 1 alone sufficient ? I mean can't fractions also be considered in statement 1 and couldnt Abs value of 1/2 = 2^-1 ? (i mean isnt that mathematical right) All i could say was that J is definitely positive .. I got 1 as well but I had this fraction option too so didn't go for Statement 1. Is it wrong to consider the fraction ? i dont know if im trying to be too analytical over this but don't want to make this mistake going into the exam Could some one pls pls help.

No, the math is not correct. If J = 1/2, then: \(|J| = \frac{1}{2}\) but \(J^{-1}=(\frac{1}{2})^{-1}=2\).

Oh man right the math is wrong, J stands for the whole fraction itself !!!! thanks a lot Bunuel, your awesome ...