Let's solve the given expression step by step, including uncertainties: Given: \[ 9.43 (\pm 0.05) \times \left[ (6.2 \pm 0.2) \times 10^{-3} + (4.1 \pm 0.1) \times 10^{-3} \right] \] --- ### **Step 1: Add the two numbers inside the bracket (considering their uncertainties)** \[ (6.2 \pm 0.2) \times 10^{-3} + (4.1 \pm 0.1) \times 10^{-3} \] Add the central values: \[ 6.2 + 4.1 = 10.3 \] Add the uncertainties (since uncertainties add for addition/subtraction): \[ 0.2 + 0.1 = 0.3 \] So, \[ (10.3 \pm 0.3) \times 10^{-3} \] **Explanation:** When adding quantities, their uncertainties are also added. --- ### **Step 2: Multiply by 9.43 (±0.05) and calculate the resulting uncertainty** Now multiply: \[ 9.43 (\pm 0.05) \times (10.3 \pm 0.3) \times 10^{-3} \] Multiply the central values: \[ 9.43 \times 10.3 = 97.129 \] Calculate the relative (fractional) uncertainties and add for multiplication: - For \(9.43\): \(\frac{0.05}{9.43} \approx 0.0053\) - For \(10.3\): \(\frac{0.3}{10.3} \approx 0.0291\) - Total relative uncertainty: \(0.0053 + 0.0291 = 0.0344\) So, the absolute uncertainty in the product is: \[ 0.0344 \times 97.129 \approx 3.34 \] So the answer is: \[ (97.1 \pm 3.3) \times 10^{-3} \] Or, writing with the same power of ten: \[ (0.097 \pm 0.003)\ (\text{rounded to 3 significant digits, matching uncertainties}) \] --- ## **Final Answer:** \[ \boxed{(0.097 \pm 0.003)} \] **All steps included calculation and propagation of uncertainties as required.**