### Solving Exponential Equations Using a Common Base #### Problem Statement: Solve the equation \(4^{x+3} = 2^{3x + 1}\) by using a common base. Enter your answer in the box provided. \[ \text{x = } \_\_\_\_\_ \] --- #### Solution Approach: 1. **Rewrite in Terms of a Common Base:** - Notice that 4 is \(2^2\). Thus, the equation \(4^{x+3}\) can be rewritten as \((2^2)^{x+3}\). - Rewrite the equation using the common base: \( (2^2)^{x+3} = 2^{3x + 1} \). 2. **Simplify the Exponents:** - Apply the power of a power property \((a^m)^n = a^{mn}\): \[ 2^{2(x+3)} = 2^{3x + 1} \] - Simplify the left side: \[ 2^{2x + 6} = 2^{3x + 1} \] 3. **Set the Exponents Equal:** - Since the bases are the same (base 2), you can set the exponents equal to each other: \[ 2x + 6 = 3x + 1 \] 4. **Solve for x:** - Isolate \(x\) by moving all \(x\)-terms to one side: \[ 2x + 6 - 3x = 1 \] \[ -x + 6 = 1 \] \[ -x = 1 - 6 \] \[ -x = -5 \] \[ x = 5 \] #### Answer: \[ x = 5 \] --- Enter your answer in the box: \[ x = 5 \] This equation solving method demonstrates how to manipulate and solve exponential equations by expressing them with a common base. Use these steps to simplify and find the solution efficiently.