The given expression can be simplified by using the formula for the sum of two cube roots:

∛(a) + ∛(b) = ∛(a+b+3√(ab))

In this case, a = 16 + 8√5 and b = 16 - 8√5.

Now, let's find the value of a + b and ab:

a + b = (16 + 8√5) + (16 - 8√5)
= 32
ab = (16 + 8√5)(16 - 8√5)
= 256 - 64(5)
= 256 - 320
= -64

Now, substitute these values into the formula:

∛(16 + 8√5) + ∛(16 - 8√5)
= ∛(32 - 64)
= ∛(-32)
= -2

Therefore, the simplified expression is -2.