\( 4.3 ^ { x } + 3 ^ { x } = 15 \)

To solve the equation: \[ 4.3^{x} + 3^{x} = 15 \] we can use numerical methods since it's not straightforward to solve algebraically. Here's a step-by-step approach to approximate the value of \( x \): ### Step 1: Understanding the Equation We need to find the value of \( x \) such that the sum of \( 4.3^x \) and \( 3^x \) equals 15. ### Step 2: Initial Estimates Start by evaluating the function at some integer values of \( x \): 1. **For \( x = 1 \):** \[ 4.3^1 + 3^1 = 4.3 + 3 = 7.3 \quad (\text{Less than } 15) \] 2. **For \( x = 2 \):** \[ 4.3^2 + 3^2 = 18.49 + 9 = 27.49 \quad (\text{Greater than } 15) \] This indicates that the solution lies between \( x = 1 \) and \( x = 2 \). ### Step 3: Narrowing Down the Value Let's try values between 1 and 2: - **For \( x = 1.5 \):** \[ 4.3^{1.5} \approx 8.91 \quad \text{and} \quad 3^{1.5} \approx 5.20 \] \[ 8.91 + 5.20 = 14.11 \quad (\text{Still less than } 15) \] - **For \( x = 1.6 \):** \[ 4.3^{1.6} \approx 10.31 \quad \text{and} \quad 3^{1.6} \approx 5.80 \] \[ 10.31 + 5.80 = 16.11 \quad (\text{Greater than } 15) \] So, the solution lies between \( x = 1.5 \) and \( x = 1.6 \). ### Step 4: Refining the Estimate Using linear approximation between \( x = 1.54 \) and \( x = 1.55 \): - **For \( x = 1.54 \):** \[ 4.3^{1.54} \approx 9.43 \quad \text{and} \quad 3^{1.54} \approx 5.45 \] \[ 9.43 + 5.45 = 14.88 \quad (\text{Close to } 15) \] - **For \( x = 1.55 \):** \[ 4.3^{1.55} \approx 9.54 \quad \text{and} \quad 3^{1.55} \approx 5.51 \] \[ 9.54 + 5.51 = 15.05 \quad (\text{Very Close to } 15) \] ### Step 5: Final Approximation Using linear interpolation between \( x = 1.54 \) and \( x = 1.55 \), the solution is approximately: \[ x \approx 1.55 \] ### Conclusion The solution to the equation \( 4.3^{x} + 3^{x} = 15 \) is approximately: \[ x \approx 1.55 \]