Solve for n over the real numbers:
2^(3 n+2) (2^n)^n = 1
Take the natural logarithm of both sides and use the identities log(a b) = log(a)+log(b) and log(a^b) = b log(a):
log(2) n^2+log(2) (3 n+2) = 0
Expand out terms of the left hand side:
log(2) n^2+3 log(2) n+2 log(2) = 0
The left hand side factors into a product with three terms:
log(2) (n+1) (n+2) = 0
Divide both sides by log(2):
(n+1) (n+2) = 0
Split into two equations:
n+1 = 0 or n+2 = 0
Subtract 1 from both sides:
n = -1 or n+2 = 0
Subtract 2 from both sides:
Answer: |n = -1 or n = -2