Math question

A negative raised to an odd power will result in a negative.

but will it always be a negative?

Proof:  [ let a > 0....so -a < 0 ....and let 2n + 1 be some odd integer]

(-a)^{2n + 1}     =

(-a)²ⁿ * (-a) =

(-1 * a)²ⁿ * (-1 * a) =

(-1)²ⁿ * (a)²ⁿ * (-1)* (a)  =

(-1²)ⁿ * (a²)ⁿ * (-a)  =

(1)ⁿ * (a²)ⁿ * (-a)  =     [1 raised to any power = 1....so we can ignore this term ]

(a²)ⁿ * (-a)    ....      and a²  for any "a" not equal to 0 is always positive....and a positive raised to any power "n" is also positive....so....the first term is always positive....and since -a < 0, the entire expression is < 0