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