Question

Explain the error in the problem worked out below. (2pts) Correct the work and show how to get the answer. (2pts) Solve the exponential equation for x: 4 ^ (2x + 1) = 8 ^ (x + 2); 4 ^ (2x + 1) = 4 ^ (2(x + 2)); 2x + 1 = 2(x + 2); 2x+1=2x+4|; - 2x; 1 ne4 no solution

Answer 1

The error in the problem lies in the manipulation of the exponential equation and the subsequent steps taken to solve for \( x \). Let's go through the correct steps to solve the equation \( 4^{(2x + 1)} = 8^{(x + 2)} \).

### Step 1: Rewrite the bases

First, we can express both sides of the equation with the same base. Note that:

- \( 4 = 2^2 \)
- \( 8 = 2^3 \)

Thus, we can rewrite the equation:

\[
4^{(2x + 1)} = (2^2)^{(2x + 1)} = 2^{2(2x + 1)} = 2^{(4x + 2)}
\]

Similarly, for the right side:

\[
8^{(x + 2)} = (2^3)^{(x + 2)} = 2^{3(x + 2)} = 2^{(3x + 6)}
\]

### Step 2: Set the exponents equal to each other

Now that both sides are expressed with the same base, we set the exponents equal to each other:

\[
4x + 2 = 3x + 6
\]

### Step 3: Solve for \( x \)

Now we'll solve for \( x \):

1. Subtract \( 3x \) from both sides:
\[
4x - 3x + 2 = 6
\]
\[
x + 2 = 6
\]

2. Subtract 2 from both sides:
\[
x = 4
\]

### Summary of the solution

The correct solution to the equation \( 4^{(2x + 1)} = 8^{(x + 2)} \) gives \( x = 4 \).

The original error was in how the last steps were manipulated; after the step \( 2x + 1 = 2(x + 2) \), it wasn't simplified correctly. Rather than leading to no solution, it leads to \( x = 4 \).

Thus, the final corrected answer is \( x = 4 \).