Uncertainty In Powers And Roots

Why powers and roots change measurement uncertainty

A small uncertainty in an input can become a different-sized uncertainty in a quantity calculated from it. Squaring a measured length, taking a cube root of a measured volume, or calculating a reciprocal power all transform the input, so the output uncertainty depends on the function as well as on the measurement.

Differentials give a first-order estimate of that change. For powers and roots, the derivative leads to a useful pattern: the predicted relative output error is a constant multiple of the relative input error. The multiplier depends on the exponent.

Power functions and the differential estimate

Let \(x_0>0\) be a measured input, and suppose its uncertainty is at most \(u\), so that an actual input \(x\) satisfies \(|x-x_0|\leq u\). Write \(y=f(x)=x^p\), where \(p\) is a fixed real number. Since \(x_0\) is positive, this power is defined and differentiable in a neighbourhood of \(x_0\).

The differential at \(x_0\) predicts the change in output when the input changes by \(\Delta x=x-x_0\). The power rule gives \(f'(x)=p x^{p-1}\), so the predicted signed change is \(dy=f'(x_0)\,dx\), with \(dx=\Delta x\). If only the size of the input uncertainty is known, use \(u\) for the maximum input-change magnitude.

The signed differential \(dy=p x_0^{p-1}dx\) also indicates the predicted direction of change. When \(p>0\), an increase in \(x\) predicts an increase in \(y\); when \(p<0\), it predicts a decrease. An uncertainty stated only as a size \(u\) has no direction, so its predicted output uncertainty is reported as a magnitude.

Why the relative-error formula has this form

The key comparison is between the size of the differential and the output value at the measured input. Dividing the differential magnitude by \(\lvert y_0\rvert\) turns the predicted absolute change into a predicted relative change. For \(p\ne0\) and \(x_0>0\), the calculation is:

Here \(x_0^p>0\), and the exponent laws give \(x_0^{p-1}/x_0^p=1/x_0\). Thus the input's relative change, \(\lvert dx\rvert/x_0\), is multiplied by \(\lvert p\rvert\). Replacing \(\lvert dx\rvert\) with the uncertainty limit \(u\) gives the stated estimate. The derivation uses the differential, so it describes the first-order predicted change rather than asserting that a finite change follows the formula exactly.

Worked Example: A fourth power

Roots are fractional powers

For a positive input, the positive \(n\)th root is \(x^{1/n}\), where \(n\) is a positive integer. It is therefore covered by the power formula with \(p=1/n\). Its predicted relative output uncertainty is the relative input uncertainty divided by \(n\).

More generally, a root can be combined with a power: \(x^{m/n}\), for integers \(m\) and positive \(n\), has exponent \(p=m/n\) on positive inputs. The relative uncertainty multiplier is \(\lvert m/n\rvert\). The absolute uncertainty still depends on the measured input through the derivative, so the relative multiplier alone does not specify the size of the output change in the units of \(y\).

Worked Example: A cube root

Negative powers and decreasing outputs

A negative exponent describes a reciprocal power. For example, \(x^{-1/2}=1/\sqrt{x}\) for \(x>0\). Its derivative is negative, so an increase in the input predicts a decrease in the output. The predicted magnitude of relative output uncertainty is still given by the absolute value of the exponent: for exponent \(-1/2\), it is half the relative input uncertainty.

Common mistake: multiplying root error by the root index

A common error is to multiply the relative input uncertainty by \(n\) when taking an \(n\)th root. The exponent of the root is \(1/n\), so its magnitude—not its reciprocal—is the multiplier for the relative output estimate.

Another mistake is to report a signed differential as an uncertainty magnitude. In the reciprocal-square-root example, \(dy=-0.001\) describes the predicted direction for an input increase of \(2\). If the question asks for the size of the uncertainty, report \(\lvert dy\rvert=0.001\), and retain the negative sign only when discussing direction.

What the estimate does—and does not—say

The derivative describes the local rate of change at the measured input. A differential estimate uses that one rate to predict the output change for a small input change. It is most useful when the uncertainty is small compared with \(x_0\), so that a first-order prediction is appropriate. The actual output change need not equal the differential exactly, because a power or root function can curve as the input changes.

These formulas also require attention to the domain. The statements here assume \(x_0>0\) and input values near \(x_0\) remain positive. This keeps real powers and roots defined in the setting used here, and avoids the singularity at zero for negative powers and roots with exponent less than one. For \(p=0\), \(x^p=1\) for positive \(x\), so the output has no change at all; the relative formula is stated separately for \(p\ne0\).

The practical advantage is that the relative estimate can be found without first calculating the absolute output uncertainty. Once the exponent is known, multiply the relative input uncertainty by its magnitude. To obtain an absolute estimate as well, use the derivative at the measured input and multiply by the input uncertainty. Keep both results tied to the measured value and describe them as first-order estimates.