What is the solution for x in the equation (2³)ⁿ * (4⁴) * (8)ⁿ = 16 * 4²?

To solve the equation ((2^3)^n \cdot (4^4) \cdot (8^n) = 16 \cdot 4^2), we first express all terms in the equation using base (2).

Starting with the left side:

• We can rewrite (4) as (2^2), so (4^4 = (2^2)^4 = 2^8).

• Also, (8) can be expressed as (2^3), so (8^n = (2^3)^n = 2^{3n}).

• Therefore, the left side becomes ((2^3)^n \cdot 2^8 \cdot 2^{3n} = 2^{3n} \cdot 2^8 \cdot 2^{3n}).

• Since the bases are the same, we can add the exponents: (3n + 8 + 3n = 6n + 8). Thus, the left side simplifies to (2^{6n + 8}).

Now for the right side:

• Here, (16) can be