Combining uncertainties in measurements that form part of a more complex formula requires a systematic approach known as the propagation of uncertainty. This approach involves differentiating the formula with respect to each measurement, multiplying each derivative by the uncertainty in that measurement, and then combining these products. The final uncertainty is typically the square root of the sum of these squared products. This method is based on Taylor series expansion and assumes that uncertainties are small and the measurements are independent. It's a fundamental technique in error analysis, allowing for a more accurate determination of uncertainty in complex calculations. It's important to note that this method assumes a normal distribution of errors and linear approximations, which may not always be valid. In such cases, more advanced statistical methods or numerical simulations might be required to accurately estimate the uncertainty.

Expressing uncertainties as a percentage is not always necessary but is often convenient for comparing and combining uncertainties in different measurements. Percentage uncertainties provide a relative sense of how significant the uncertainty is in relation to the measurement itself. This is particularly useful when dealing with measurements of different magnitudes or units, as it allows for a standardized comparison. However, in some cases, especially when dealing with direct measurements or when reporting final results, it might be more appropriate to express uncertainty in absolute terms. The choice between percentage and absolute uncertainty often depends on the context and the specific requirements of the experiment or calculation. In academic and professional settings, the standard practice or guidelines provided should be followed. Regardless of the form, the key is to maintain consistency and clarity in how uncertainties are presented and interpreted.

Uncertainties are inherently non-negative, as they represent the range within which the true value of a measurement is expected to lie. They indicate the magnitude of deviation from the measured value, not the direction. Therefore, uncertainties are always expressed as positive values, whether they are absolute (e.g., ±0.2 cm) or relative (e.g., ±2%). In calculations, uncertainties are treated mathematically as absolute magnitudes. For instance, when adding or subtracting measurements, their absolute uncertainties are added, regardless of whether the measurements themselves are being added or subtracted. Similarly, in multiplication, division, or when raising to a power, the relative (percentage) uncertainties are combined in a way that reflects the overall uncertainty of the result without considering the sign of the original measurements. This approach ensures that the uncertainty reflects a range of possible values in which the true value is likely to lie.

When dealing with measurements that are squared or cubed, the approach to handling uncertainties changes. For a measurement raised to a power (n), the relative uncertainty of the measurement is multiplied by the absolute value of that power. This rule stems from the principles of propagation of uncertainties in mathematical operations. For example, if a measurement A has a relative uncertainty and is squared (A²), the relative uncertainty of A² is 2 times the relative uncertainty of A. Similarly, if A is cubed (A³), the relative uncertainty of A³ is 3 times the relative uncertainty of A. This multiplication factor reflects how the uncertainty grows more rapidly than the measurement itself as it's raised to higher powers. It's important to note that this rule applies only to relative uncertainties; absolute uncertainties require different methods of calculation that consider the specific nature of the mathematical operation involved.

When combining measurements with different degrees of uncertainty, it's essential to consider the relative impact of each uncertainty on the final result. The general rule is to use the largest relative uncertainty when the measurements contribute equally to the final value. This is because the most significant uncertainty has the greatest potential to affect the overall accuracy of the result. However, if one measurement significantly outweighs others in terms of its contribution to the final value, the uncertainty associated with this dominant measurement should be prioritized. For instance, in a multiplication involving a very large number with a small uncertainty and a small number with a large uncertainty, the larger relative uncertainty of the smaller number might have a minimal impact on the final result compared to the smaller relative uncertainty of the larger number. It's crucial to assess each situation on a case-by-case basis, evaluating the contribution of each measurement to the final value and its associated uncertainty.