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)