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The following link How to determine significant figures involving radicals and exponents mentions that if you have 5.1^4 "The 4 is (probably) exact, so we ignore that for deciding precision, so the answer should have two significant digits, just as 5.1 does. That give an answer of 680". In the above quoted example it appears the base is considered a measured value and the exponent an exact value.

What is the rule for when both the base and exponent are considered measured values. Consider the following power where the base and exponent are considered measured quantities: 3.21^2.33

What about when the base is an exact value and the exponent is a measured value?

van fifty
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  • Significant figures are questionable when raising to high powers because the value can differ so much. In your example of $5.1^4$ the real value could range from $5.05^4 \approx 650$ to $5.15^4 \approx 703$ That looks like only one significant figure to me. They are a "quick and dirty" way of doing error propagation. – Ross Millikan Jan 28 '21 at 01:34

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All in all I think you're trying to push significant figures beyond their normal use. The gist is that significant figures are a very simple method to error propagation. The basic interpretation is to essentially think of the values being used as rounded. So $x= 3.21$ implies that $x$ should be in the range $3.205 < x < 3.215$.

In the case $3.21^{2.33}$ it is assumed that there are errors in both values so:

$$3.215^{2.335} = 15.29$$ $$3.21^{2.33} = 15.14$$ $$3.205^{2.325} = 15.00$$

So the high difference is 0.15 and the low difference is 0.14. Take the bigger value and you get 15.14(15) indicating that the error in the last two digits is 15, so $14.99 \le \mathrm{value} \le 15.29$.

Now there is a bit of a further dilemma. Do you want to express just the value in significant figures, or are you satisfied with the odd 15.14(15) which is common in scientific writings? If you want to stick with just the value in significant figures then I'd use 15.1 for the value. Having 0.05 for the error on 15.1 loses some information, but it keeps you from foolishly proposing that the value should be "approximately" $15.141 075 516 144 591 945 214 121 498 099$ implying an error of $\pm 0.000 000 000 000 000 000 000 000 000 000 5$.

MaxW
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