Solve the following expressions .
`4^(2x + 1) = 8^(x + 3)`, then x = ?

Answer

To solve the equation \( 4^{(2x + 1)} = 8^{(x + 3)} \), we will follow these steps:

### Step 1: Rewrite the bases
First, we need to express both sides of the equation with the same base. We know that:
- \( 4 = 2^2 \)
- \( 8 = 2^3 \)

So we can rewrite the equation as:
\[ (2^2)^{(2x + 1)} = (2^3)^{(x + 3)} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 2^{2(2x + 1)} = 2^{3(x + 3)} \]

This simplifies to:
\[ 2^{(4x + 2)} = 2^{(3x + 9)} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 4x + 2 = 3x + 9 \]

### Step 4: Solve for \( x \)
Now, we will isolate \( x \):
1. Subtract \( 3x \) from both sides:
\[ 4x - 3x + 2 = 9 \]
This simplifies to:
\[ x + 2 = 9 \]

2. Subtract \( 2 \) from both sides:
\[ x = 9 - 2 \]
Thus, we find:
\[ x = 7 \]

### Final Answer
The value of \( x \) is \( 7 \).

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