How many significant digits should be retained on powers?

Question

An example:

$$5.1^4=676.5201$$

But the base of the power only has two significant figures.

• In this case, How this result should be reported?

Answer 1

A quick hands on rule is to try the worst cases, and use some good sense.

So, $$X=5.1^4=676.5201$$

Assuming $4$ is a exact number and there is some uncertainty about $5.1$, then the 'real' result will be between $$X_{min}=5.05^4\approx650.3775\dots$$ and $$X_{max}=5.15^4\approx703.4430\dots$$

So the second digit of the result is uncertain but it still looks like a fair guess. Also note that the mid point is roughly 30 units away from the ends. Two possible representations would be:

$$X\approx680\pm30$$ $$X\approx6.8\times10^2$$