To simplify the given expression, let's start by analyzing it:

$\frac{(2 \cdot 3^{x+1})^3}{\sqrt[3]{3^{16x} \cdot 12^{3 - 2x}}}$

Step 1: Expand the Numerator

The numerator is $(2 \cdot 3^{x+1})^3$. We can expand this by distributing the exponent: $(2 \cdot 3^{x+1})^3 = 2^3 \cdot (3^{x+1})^3 = 8 \cdot 3^{3(x+1)} = 8 \cdot 3^{3x + 3}$

Step 2: Simplify the Denominator

The denominator is the cube root of $3^{16x} \cdot 12^{3 - 2x}$: $\sqrt[3]{3^{16x} \cdot 12^{3 - 2x}}$

Step 2.1: Simplify $12^{3 - 2x}$

We know that $12 = 2^2 \cdot 3$, so: $12^{3 - 2x} = (2^2 \cdot 3)^{3 - 2x} = 2^{2(3 - 2x)} \cdot 3^{3 - 2x} = 2^{6 - 4x} \cdot 3^{3 - 2x}$

Now we substitute this back into the denominator: $\sqrt[3]{3^{16x} \cdot 2^{6 - 4x} \cdot 3^{3 - 2x}}$

Step 2.2: Combine the $3$-Terms in the Denominator

Now we combine the powers of $3$ in the denominator: [ \sqrt[3]{3^{16x + 3 - 2x} \cdot 2^{6 - 4x}} = \sqrt[3]{3^{14x + 3} \cdot 2^{6 - 4x}} ]

Step 2.3: Apply the Cube Root

Applying the cube root to each term: $\sqrt[3]{3^{14x + 3} \cdot 2^{6 - 4x}} = 3^{\frac{14x + 3}{3}} \cdot 2^{\frac{6 - 4x}{3}}$

Step 3: Write the Expression in Terms of Simplified Powers

We now have: $\frac{8 \cdot 3^{3x + 3}}{3^{\frac{14x + 3}{3}} \cdot 2^{\frac{6 - 4x}{3}}}$

Step 4: Simplify the Fraction

Separate the powers of $2$ and $3$:

1. For $2$: $\frac{8}{2^{\frac{6 - 4x}{3}}} = 2^{3 - \frac{6 - 4x}{3}} = 2^{3 - 2 + \frac{4x}{3}} = 2^{1 + \frac{4x}{3}}$

2. For $3$: $\frac{3^{3x + 3}}{3^{\frac{14x + 3}{3}}} = 3^{3x + 3 - \frac{14x + 3}{3}}$