To solve the problem, we need to evaluate the expression \(4 * (8 * 32)\) using the operation defined as \(P * Q = \sqrt{PQ}\). Let's break this down step-by-step. ### Step 1: Evaluate the inner operation \(8 * 32\) According to the operation defined, we have: \[ 8 * 32 = \sqrt{8 \cdot 32} \] Now, we need to calculate \(8 \cdot 32\): \[ 8 \cdot 32 = 256 \] Next, we take the square root of 256: \[ \sqrt{256} = 16 \] So, we find that: \[ 8 * 32 = 16 \] ### Step 2: Substitute back into the original expression Now that we have evaluated \(8 * 32\), we can substitute this value back into the original expression: \[ 4 * (8 * 32) = 4 * 16 \] ### Step 3: Evaluate the operation \(4 * 16\) Using the same operation definition, we calculate: \[ 4 * 16 = \sqrt{4 \cdot 16} \] Now, we calculate \(4 \cdot 16\): \[ 4 \cdot 16 = 64 \] Next, we take the square root of 64: \[ \sqrt{64} = 8 \] ### Final Answer Thus, the value of \(4 * (8 * 32)\) is: \[ \boxed{8} \] ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option B: 3** - This value does not relate to any calculations we performed. The operations we executed led us to 8, and there is no mathematical basis for arriving at 3 in this context. - **Option C: 16** - This option might seem plausible because we calculated \(8 * 32\) to be 16. However, we need to remember that we are looking for \(4 * (8 * 32)\), which we found to be 8, not 16. - **Option D: 4** - This option is also incorrect. While 4 is a number involved in our calculations, it does not represent the final result of the operation we performed. ### Summary of Key Points - The operation defined is \(P * Q = \sqrt{PQ}\). - We first evaluated \(8 * 32\) to get 16. - We then evaluated \(4 * 16\) to get 8. - The correct answer is \(8\), which corresponds to option A. ### Revision Summary - Understand the operation defined: \(P * Q = \sqrt{PQ}\). - Break down complex expressions into simpler parts. - Always substitute back the results into the original expression. - Double-check calculations to avoid common pitfalls, such as misinterpreting the order of operations.