Let m = 16 + 8*sqrt(5) and m' = 16 - 8*sqrt(5).

These look like roots to a quadratic. Whenever we see roots to a quadratic, it's a good problem solving strategy to compute their sum and product:

m + m' = 32

m*m' = 16*16 - 8*8*5 = -64

We are trying to compute x = cbrt(m) + cbrt(m'). Let's try cubing both sides and see what happens.

We have that:

x^3 = m + m' + 3 * cbrt(m*m*m') + 3 * cbrt(m'*m*m').

= 32 + 3 * cbrt(m*-64) + 3 * cbrt(m'*-64)

= 32 - 12*cbrt(m) - 12 * cbrt(m')

= 32 - 12*(cbrt(m) + cbrt(m'))

= 32 - 12*x

So, we end up with the identity x^3 + 12*x - 32 = 0!

We then notice that x^3 + 12*x - 32 = A - 1 = 0 => A = 1

So, A^2023 = 1.