Seloct the correct location on the solution path. Sean was given the following exponential equation to solve. \[ \begin{aligned} 7^{(2 x-2)} & =49^{(3 x+2)} \\ 7^{(2 x-2)} & =49^{(3 x+2)} \\ 7^{(2 x-2)} & =\left(7^{2}\right)^{(3 x+2)} \\ 7^{(2 x-2)} & =7^{(6 x+2)} \\ 2 x-2 & =6 x+2 \\ -4 x-2 & =2 \\ -4 x & =4 \\ x & =-1\end{aligned} \]

It looks like there's an error in the solution path provided for Sean's exponential equation. Let's walk through the steps to identify and correct the mistake. ### Given Equation: \[ 7^{(2x - 2)} = 49^{(3x + 2)} \] ### Step-by-Step Solution: 1. **Express Both Sides with the Same Base:** - Since \(49 = 7^2\), we can rewrite the equation as: \[ 7^{(2x - 2)} = (7^2)^{(3x + 2)} \] 2. **Apply the Power of a Power Property:** - \((a^m)^n = a^{m \cdot n}\), so: \[ 7^{(2x - 2)} = 7^{2 \cdot (3x + 2)} \] \[ 7^{(2x - 2)} = 7^{6x + 4} \] **_Error Identified Here:_** - In the original solution, this step was incorrectly simplified to \(7^{6x + 2}\) instead of \(7^{6x + 4}\). 3. **Set the Exponents Equal to Each Other:** \[ 2x - 2 = 6x + 4 \] 4. **Solve for \(x\):** \[ 2x - 6x = 4 + 2 \] \[ -4x = 6 \] \[ x = -\frac{6}{4} \] \[ x = -\frac{3}{2} \] ### Correct Solution: \[ x = -\frac{3}{2} \] ### Summary of the Correction: - **Incorrect Step:** \[ 7^{(2x - 2)} = 7^{6x + 2} \] - **Corrected Step:** \[ 7^{(2x - 2)} = 7^{6x + 4} \] By correcting the exponent from \(6x + 2\) to \(6x + 4\), the final solution for \(x\) becomes \(-\frac{3}{2}\) instead of \(-1\).