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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?
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    What is the rule for multiplication? – 2017-01-27
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    Depends on what you know about that $5.1$. If this were a grade school assignment, then a student might be expected to give all the digits. The same applies if $5.1$ is known to be an exact value. OTOH, if it is a measurement, then whoever did the measuring is supposed to give an estimate of the margin of error, or a confidence interval. Also, the answer to your question will depend on how many significant digits are useful to you (or whoever will be using your figures to their ends). For example, if that $5.1$ has a relative inaccuracy of $\pm 1\%$, then $5.1^4$ only known upto $\pm 4\%$ – 2017-01-27
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    @JyrkiLahtonen What would be the answer for each scenario? Having in mind the **arithmetic power** – 2017-01-27
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    You are probably expected to retain two digits, as that's what you were given. – 2017-01-27
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    @Kaynex Yes.... Following your words, the result would be 67.... but what happen with the rest? – 2017-01-27
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    No the result would not be 67 but 670 or using scientific notation $6.7*10^2$ – 2017-01-27
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    It also wouldn't be 670 as it rounds to 680. – 2017-01-27

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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:

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