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Error in powers
Intermediate
Relative error is multiplied by the exponent.
Δz/z = n·ΔA/AIf Z = Aⁿ the relative error is multiplied by the exponent: ΔZ/Z = n·ΔA/A. For several factors, each term contributes |exponent| times its relative error, so quantities raised to high powers dominate the uncertainty.
A sphere radius measured to 1% gives volume (r³) with 3% error and area (r²) with 2% error.
The exponent can be negative or fractional, but the contribution to the error is always added, using its absolute value.
In more detail
For a quantity raised to a power n, the relative error is multiplied by the magnitude of n. So the relative error in the area of a circle, which depends on r squared, is twice the relative error in r.
Real-world uses
- A 1% error in the radius of a sphere gives about a 3% error in its volume, since volume depends on r cubed.
- Calculating kinetic energy from speed, which is squared, doubles the relative error in speed.
- The area of a circle depends on the square of its radius, so a 2 percent error in the radius gives about a 4 percent error in the area.
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Frequently asked questions
Why is the power rule useful?
It lets you estimate how errors grow in formulas such as areas, volumes and energies without a full calculation.
Does a negative power change the rule?
No. The error is multiplied by the magnitude of the power, so an inverse-square law still magnifies relative errors.
Does the rule work for square roots?
Yes. A square root is a power of one half, so the relative error is halved.
Related topics in Measurement & Math Tools
Relative and percent error · Error in a sum or difference · Error in product or quotient · Mean and standard deviation · Dimensions of a quantity · Dimensional homogeneity