Here's a link to the ways to calculate distances of various objects. Many of the earlier proofs (like heliocentrism) can be proved by experiments that are within your capacity.

Here's a list of various putative phenomena. Many, like astrology, don't work. Some, like quantum electrodynamics, do work, as shown by the fact that computers work.
So, there are practical and verifiable differences in the world based on the truth or falsity of predictive theories.

But you cannot deny that there is a huge body of information we accept to be truth soley based on the authority that provides it.

In practice, almost all information (schooling, etc.). So what? Information that we learn from scientific (i.e. accurate prediction) processes is universalizable, at least to the extent that the scientist complies with the scientific rules. (That rules out Lysenkoism as universalizable). That's the point of the examples that I listed.
Experts say that GPS works because relativity is true, and GPS works. If you start analyzing relativity using power relations, you can question GPS or question the veracity of the experts. But GPS manifestly works. So, suspect the experts. But suspect them of what? Providing technology that works? They don't deny. Using magic? Is that really the best explanation?

pre-legitimized mathematical formula.

What?!? Mathematics is non-empirical. If you are unsure whether 2 + 3 = 5 based on power relations, how do you explain the consistency of reality? Power relations are the method of analyzing moral truths. I accept that the line between moral and scientific truths is sometimes blurry, but there is a difference between those categories.