To solve the equation \( 2^{x+3} = 32 \), we need to find the value of \( x \). Let's break this down step-by-step. ### Step 1: Understand the Equation The equation \( 2^{x+3} = 32 \) involves an exponent. The left side has a base of 2 raised to the power of \( x + 3 \), and the right side is the number 32. ### Step 2: Express 32 as a Power of 2 To solve for \( x \), it helps to express both sides of the equation with the same base. We know that: \[ 32 = 2^5 \] This means we can rewrite the equation as: \[ 2^{x+3} = 2^5 \] ### Step 3: Set the Exponents Equal Since the bases are the same (both are base 2), we can set the exponents equal to each other: \[ x + 3 = 5 \] ### Step 4: Solve for \( x \) Now, we can solve for \( x \) by isolating it: \[ x + 3 - 3 = 5 - 3 \] \[ x = 2 \] ### Conclusion The value of \( x \) is \( 2 \). Therefore, the correct option is **A**. ### Explanation of Other Options - **Option B (3)**: This is incorrect because if \( x = 3 \), then \( x + 3 = 6 \) and \( 2^6 = 64 \), which does not equal 32. - **Option C (5)**: This is incorrect because if \( x = 5 \), then \( x + 3 = 8 \) and \( 2^8 = 256 \), which does not equal 32. - **Option D (7)**: This is incorrect because if \( x = 7 \), then \( x + 3 = 10 \) and \( 2^{10} = 1024 \), which does not equal 32. ### Common Pitfalls - **Misunderstanding Exponents**: Students often confuse the operations with exponents. Remember that when the bases are the same, you can set the exponents equal. - **Forgetting to Simplify**: Always simplify your equations step-by-step to avoid mistakes. ### Revision Summary - Convert numbers to the same base when dealing with exponential equations. - Set the exponents equal when the bases are the same. - Solve for the variable by isolating it. - Check your work by substituting back into the original equation. By following these steps, you can confidently solve similar exponential equations in the future!