Multiply and Simplify:

Question

(a^x-a^(-x))^3
=(a^x-a^(-x))(a^x-a^(-x))(a^x-a^(-x))
=(a^x-a^(-x))(a^2x-a^0-a^0+a^(-2x))
=(a^x-a^(-x))(a^2x-2+a^(-2x))
=a^3x-2a^x+a^(-x)-a^x+2a^(-x)-a^(-3x)
=a^3x-3a^x+3a^(-x)-a^(-3x)
=???

Can you simplify down any further?

please and thank you

Answer 1

using the general
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

(a^x - a^-x)^3
= a^3x - 3a^2xa^-x + 3a^xa^-2x - a^-3x
= a^3x - 3a^x + 3a^-x - a^-3x

Not sure where you want to go from here.

Answer 2

It would be good as it is. If you want, you can replace
3a^(-x) by 3/a^x and
-a^(-3x) by -1/a^(3x)
which may be considered more explicit.