# How many significant digits should be retained on powers?

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?
• What is the rule for multiplication? – scrappedcola Jan 27 '17 at 19:31
• 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 $\pm 4\%$ – Jyrki Lahtonen Jan 27 '17 at 19:36
• @JyrkiLahtonen What would be the answer for each scenario? Having in mind the arithmetic power – Another.Chemist Jan 27 '17 at 19:40
• You are probably expected to retain two digits, as that's what you were given. – Kaynex Jan 27 '17 at 19:40
• @Kaynex Yes.... Following your words, the result would be 67.... but what happen with the rest? – Another.Chemist Jan 27 '17 at 19:55

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 $$X_{max}=5.15^4\approx703.4430\dots$$
$$X\approx680\pm30$$ $$X\approx6.8\times10^2$$