To simplify the expression \(\sqrt{\frac{(0.0023 \times 750)}{(0.00345 \times 1.25)}}\), we will break it down step-by-step. ### Step 1: Calculate the Numerator and Denominator First, we need to calculate the values in the numerator and the denominator separately. **Numerator:** \[ 0.0023 \times 750 \] Calculating this: \[ 0.0023 \times 750 = 1.725 \] **Denominator:** \[ 0.00345 \times 1.25 \] Calculating this: \[ 0.00345 \times 1.25 = 0.0043125 \] ### Step 2: Form the Fraction Now we can form the fraction: \[ \frac{1.725}{0.0043125} \] ### Step 3: Perform the Division Next, we will divide the numerator by the denominator: \[ \frac{1.725}{0.0043125} = 400 \] ### Step 4: Take the Square Root Now we take the square root of the result: \[ \sqrt{400} = 20 \] ### Final Answer Thus, the simplified value of \(\sqrt{\frac{(0.0023 \times 750)}{(0.00345 \times 1.25)}}\) is: \[ \boxed{20} \] ### Explanation of Why the Answer is Correct 1. **Numerical Calculation**: We accurately calculated both the numerator and denominator, ensuring that we used the correct multiplication and division. 2. **Square Root**: We correctly applied the square root to the result of the division, which is a fundamental operation in mathematics. ### Explanation of Why Other Options are Incorrect - **Option A (15)**: This is incorrect because the calculations show that the square root of the fraction equals 20, not 15. - **Option C (40)**: This option is also incorrect as it does not match the calculated square root of 400. - **Option D (75)**: This option is incorrect for the same reason; the calculations do not support this value. ### Common Pitfalls - **Miscalculating the Multiplication**: Ensure that you multiply the numbers correctly, especially when dealing with decimals. - **Division Errors**: Be careful when dividing, as small errors can lead to large discrepancies in the final answer. - **Square Root Misapplication**: Always ensure that you are taking the square root of the final result, not intermediate steps. ### Revision Summary - Break down complex expressions into manageable parts (numerator and denominator). - Perform calculations step-by-step to avoid errors. - Always double-check your arithmetic operations. - Understand the properties of square roots to simplify expressions correctly. By following these steps, you can confidently simplify similar expressions in the future!