Solve for x when : `6^(2x+4)= 3^(3x)*2^(x+8)`.

Answer

To solve the equation \( 6^{2x+4} = 3^{3x} \cdot 2^{x+8} \), we will follow these steps:

### Step 1: Rewrite the bases
First, we can express \( 6 \) as \( 2 \cdot 3 \):
\[ 6^{2x+4} = (2 \cdot 3)^{2x+4} \]
Using the property of exponents \( (pq)^n = p^n \cdot q^n \), we can rewrite this as:
\[ 6^{2x+4} = 2^{2x+4} \cdot 3^{2x+4} \]

### Step 2: Rewrite the equation
Now, substituting this back into the original equation gives us:
\[ 2^{2x+4} \cdot 3^{2x+4} = 3^{3x} \cdot 2^{x+8} \]

### Step 3: Compare the exponents of like bases
From the equation, we can compare the exponents of \( 2 \) and \( 3 \) on both sides.

1. For the base \( 2 \):
\[ 2x + 4 = x + 8 \]

2. For the base \( 3 \):
\[ 2x + 4 = 3x \]

### Step 4: Solve the first equation
Let's solve the first equation \( 2x + 4 = x + 8 \):
\[ 2x - x = 8 - 4 \]
\[ x = 4 \]

### Step 5: Solve the second equation
Now, let's solve the second equation \( 2x + 4 = 3x \):
\[ 2x + 4 = 3x \]
\[ 4 = 3x - 2x \]
\[ x = 4 \]

### Conclusion
Both equations give us the same solution. Therefore, the solution for \( x \) is:
\[ \boxed{4} \]