If J ≠  0, what is the value of J? GMAT Data Sufficiency

It is given that J ≠ 0
S1:
|J| =\( J^{-1}\)

|J| = 1/J … (1)
It is not given that J is positive or negative
So, J can take any value.
But the condition 1 states that J is positive

From equation (1),
|J| is always positive
So, RHS has to be positive and hence J is positive.

Therefore, |J| cannot be – J
|J| = 1/J
|J| * J = 1
J = 1
Sufficient

S2:

\(J^J\)=1
J = 1
Sufficient

Correct option: D

J ≠ 0

Statement 1:

If J = 0 then we will have \( 0^{-1}\)=1/0 = undefined
Remember you can’t raise 0 to a negative power

Statement 2:
If J = 0 then we will have \(0^0\).\(0^0\), in some sources equal to 1, it is said that this is undefined.
Moving back to the question:
S1: |J| =\(J^{-1}\)

|J| * J = 1

J = 1
Here J can have no way to be a negative number, since in this case we would have |J| * J = positive * negative = negative ≠ 1
Hence, sufficient

S2: \(J^J\)=1
Again only one solution: J = 1
Hence, sufficient

Correct option: D

It should be noted that it is important to state that J ≠ 0 because in condition
1/0 could not be defined.
Also in case 2, 0 could not be raised to the power 0.
It is given in the question that J ≠ 0, so we do not have to handle that case.

Condition 1:

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

|J| = 1/J
|J| * J = 1
Here the value of J can be 1 only
Taking J = -1, the value of LHS will be -1
J cannot be a negative number as modulus is always positive, so the product of negative and positive will not give a positive number.

Hence sufficient

Condition 2:

\(J^J\)=1

J = \(1^{1/J}\)

J = 1 ( it is known that 1 raised to the power any number is always 1)
Here, we get J = 1
Hence sufficient

Correct option: D