Using this law, we can easily compare the life expectancy for different animals. For example, let's calculate the average life span of an elephant. The average weight of a male elephant is 6,000 kg ± 1,000 kg. 6,000,000 (we converted kg to gr) raised to the one-quarter power is 49.5. Thus the average life span of an elephant is 49.5 years.

African Elephant. Image: Courtesy of African Wildlife Foundation.

What should we do with the error? Raising to a power is related to products. For example, the power of 2 is nothing more than taking a product of a number with itself, y × y. We already know the rule for products − add relative errors² − so the resulting relative error for y × y is two times the relative error of y. Similarly, for other powers 3, 4, 5, ... the relative error of the result is the relative error of the original quantity times the power to which it is raised. What about fractional powers like 1/2? Well, 1/2 is the square root, which is the reverse of squaring, so the relative error calculation should also be reversed. In the case of the squaring, we multiplied the relative error by two. In the case of the square root, we should divide the relative error by two, which is the same as multiplying it by 1/2. Similarly, for other fractional powers 1/3, 1/4, ... we simply multiply the relative error by the power. So no matter what the power is, fractional or not, the rule is always the same: the relative error of the result is the relative error of the original quantity times the power.

Back to our elephant example. The relative error for the elephant mass is 17%. So the relative error for the life span is 1/4 × 17% = 4%. Therefore, our final estimate for the average life expectancy of elephants is 50 years ± 2 years.