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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 \(3\):**
   \[
   \frac{3^{3x + 3}}{3^{\frac{14x + 3}{3}}} = 3^{3x + 3 - \frac{14x + 3}{3}}
   \]



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

Learn how to fully simplify the expression \((2 \cdot 3^{x+1})^3 / \sqrt[3]{3^{16x} \cdot 12^{3 - 2x}}\) by applying laws of exponents and roots. This solution is ideal for students in Grades 10-12.

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