To solve the equation \( 3 + 4^{(2x + 8)} = 11 \) for \( x \), follow these steps:

### Step 1: Isolate the exponential term
First, subtract 3 from both sides of the equation:
\[
4^{(2x + 8)} = 11 - 3
\]
\[
4^{(2x + 8)} = 8
\]

### Step 2: Express 8 as a power of 4
Since \( 8 \) is not a direct power of 4, let's express 8 as a power of 4:
\[
4^{(2x + 8)} = 8
\]

We know that \( 8 = 2^3 \). Since \( 4 = 2^2 \), we can rewrite the equation as:
\[
(2^2)^{(2x + 8)} = 2^3
\]
Simplify the left side:
\[
2^{2(2x + 8)} = 2^3
\]

### Step 3: Equate the exponents
Since the bases are the same (both are powers of 2), equate the exponents:
\[
2(2x + 8) = 3
\]

### Step 4: Solve for \( x \)
Now, solve the equation for \( x \):
\[
4x + 16 = 3
\]
Subtract 16 from both sides:
\[
4x = 3 - 16
\]
\[
4x = -13
\]
Divide by 4:
\[
x = \frac{-13}{4}
\]
\[
x = -3.25
\]

### Final Answer
The solution to the equation \( 3 + 4^{(2x + 8)} = 11 \) is:
\[
x = -3.25
\]

Would you like more details or have any questions? Here are some related problems to consider:

1. Solve \( 4^{(2x + 8)} = 16 \).
2. Solve \( 4^{2x} + 8 = 12 \).
3. Solve \( 5 + 4^{(2x + 8)} = 20 \).
4. Solve \( 4^{(2x + 8)} - 4 = 0 \).
5. Solve \( 4^{2x} = 2^6 \).
6. Determine the solution for \( 3 + 3^{(2x + 1)} = 30 \).
7. Solve \( 3 + 2^{(2x + 8)} = 19 \).
8. Explore the function \( f(x) = 4^{(2x + 8)} + 3 \) and its properties.

**Tip:** When solving exponential equations, try to express both sides with the same base to simplify the equation.

Learn how to solve the equation 3 + 4^(2x + 8) = 11 with detailed step-by-step solutions. This problem involves exponential equations and the application of the power rule. Suitable for high school students in Grades 9-12.

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