Multiply and Simplify:

(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

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.

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.

To simplify the expression (a^x-a^(-x))^3 further, we can distribute the exponent of 3 to each term inside the parentheses:

(a^x-a^(-x))^3 = (a^x-a^(-x))(a^x-a^(-x))(a^x-a^(-x))

Now, distribute the exponent of 3 to each term inside the parentheses:

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

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

Next, simplify each term:

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

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

Since a^0 = 1, we can simplify further:

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

Now, let's simplify the term 3a^x(a^(-2x)):

= 3a^xa^(-2x)

= 3a^(x - 2x)

= 3a^(-x)

Finally, substitute this result back into the expression:

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

Therefore, the simplified form of the expression (a^x-a^(-x))^3 is:

a^3x - 3 + 3a^(-x) - a^(-3x)