To solve the equation \( 2^{3x} = 32 \), we need to find the value of \( x \). Let's break this down step-by-step. ### Step 1: Rewrite the Right Side of the Equation First, we can express \( 32 \) as a power of \( 2 \). We know that: \[ 32 = 2^5 \] So, we can rewrite the equation as: \[ 2^{3x} = 2^5 \] ### Step 2: Set the Exponents Equal Since the bases are the same (both are base \( 2 \)), we can set the exponents equal to each other: \[ 3x = 5 \] ### Step 3: Solve for \( x \) Now, we need to isolate \( x \). To do this, we divide both sides of the equation by \( 3 \): \[ x = \frac{5}{3} \] ### Step 4: Check the Options Now, let's compare our answer \( \frac{5}{3} \) with the provided options: - A. 2 - B. 3 - C. 4 - D. 5 None of these options match \( \frac{5}{3} \). It seems there was a misunderstanding in the recorded correct option. The correct answer is not listed among the options provided. ### Step 5: Verify the Calculation To ensure our solution is correct, we can substitute \( x = \frac{5}{3} \) back into the original equation: \[ 2^{3(\frac{5}{3})} = 2^5 = 32 \] This confirms that our solution is indeed correct. ### Explanation of Incorrect Options - **Option A (2)**: If \( x = 2 \), then \( 3x = 6 \) and \( 2^{3x} = 2^6 = 64 \), which does not equal \( 32 \). - **Option B (3)**: If \( x = 3 \), then \( 3x = 9 \) and \( 2^{3x} = 2^9 = 512 \), which does not equal \( 32 \). - **Option C (4)**: If \( x = 4 \), then \( 3x = 12 \) and \( 2^{3x} = 2^{12} = 4096 \), which does not equal \( 32 \). - **Option D (5)**: If \( x = 5 \), then \( 3x = 15 \) and \( 2^{3x} = 2^{15} = 32768 \), which does not equal \( 32 \). ### Summary - The equation \( 2^{3x} = 32 \) can be solved by rewriting \( 32 \) as \( 2^5 \). - Setting the exponents equal gives \( 3x = 5 \), leading to \( x = \frac{5}{3} \). - None of the provided options (A, B, C, D) are correct as they do not match \( \frac{5}{3} \). - Always verify your solution by substituting back into the original equation. ### Revision Points - Understand how to express numbers as powers of their bases. - When bases are the same, set the exponents equal to solve for the variable. - Always check your solution by substituting back into the original equation. - Be cautious of the options provided; they may not always include the correct answer.