A

Step 1: Identify the values needed to calculate the gravitational force using Newton's Law of Universal Gravitation. We have:
- Mass of the brain, m = 1.4 kg
- Mass of the moon, M = 7.34 × 10^{22} kg
- Distance, r = 378000 km = 378000 × 10^3 m = 3.78 × 10^{8} m.

Step 2: Use the formula for gravitational force:
$$ F = G \frac{m \cdot M}{r^2} $$
where G is the gravitational constant, approximately $ 6.674 \times 10^{-11} \ ext{N m}^2/\text{kg}^2 $.

Step 3: Substitute the values into the formula:
$$ F = 6.674 \times 10^{-11} \frac{1.4 \times 7.34 \times 10^{22}}{(3.78 \times 10^{8})^2} $$

Step 4: Calculate the denominator:
$$ (3.78 \times 10^{8})^2 = 1.42884 \times 10^{17} $$

Step 5: Substitute it back into the equation:
$$ F = 6.674 \times 10^{-11} \frac{1.4 \times 7.34 \times 10^{22}}{1.42884 \times 10^{17}} $$

Step 6: Calculate the numerator:
$$ 1.4 \times 7.34 \times 10^{22} = 10.276 \times 10^{22} \text{ kg m}^2/\text{s}^2 $$

Step 7: Plugging everything into the final equation gives:
$$ F = 6.674 \times 10^{-11} \frac{10.276 \times 10^{22}}{1.42884 \times 10^{17}} \approx 0.469 \text{ N} $$

Therefore, the force exerted by the moon on the brain is approximately 0.469 N, rounded off appropriately. Thus, the full answer is: F \approx 0.47 N.