I think you are right sir even though it is reported as D

manipulating

\(\frac{1}{-1^x}=-1\)

\(1=-(-1^x)\)

\(1=1^x\)

\(x=1\)

B is greater than A

Answer should be D. Why does it show B?

On simplifying the left equation

(-1)^-x=-1
=> 1/((-1)^x) = -1
=> Which clearly means x is an odd number

And since x > 0 , it can be either of 1, 3, 5, 7 and on substituting the equation hold true, hence D

The original answer was D but to have -1 on RHS when we do have 1^x , x must be zero or 1

But the stem says that x is different from zero. X must be 1 then

Will see if BrushMyQuant has something to add

Given that \((-1)^{-x}=-1\) and 𝑥 is a positive integer ≠ 0

Quantity A = x

Now, \((-1)^{-x}=-1\)

For this to be true x should be an odd integer, as if x is even then \((-1)^{-x}=1\)
Ex: \((-1)^{-1}\)=\(\frac{1}{-1}\) = -1
Ex: \((-1)^{-2}\)=\(\frac{1}{(-1)^2}\) = 1
Ex: \((-1)^{-5}\)=\(\frac{1}{(-1)^5}\) = -1

Clearly, Quantity A can be greater than or less than or equal to Quantity B also.

So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Exponents

Where is X! given in the question?

\(1=-(-1^x)\) = -(\((-1)^x\))
=> \((-1)^x\) = -1

Now, above statement will be true only for odd values of x
=> x can be 1, 3, 5...
=> Quantity A(x) can be smaller, equal to greater than Quantity B(3)

Hope it helps!

Carcass : Updated OA to D