I did, through graduate-level courses while studying for a Ph.D. in engineering - but I don't think that's meaningful here. My math instructors weren't chosen for their knowledge of English. Some of them weren't even native speakers. They knew more than I ever will about where exponents come from, but as a group they were not good role models for the right way to say them....But I didn't study mathematics at university!
I agree, though one also hears "five times ten to the five" (though because of the repetition of the "five," that would sound a little awkward, and "fifth" would be more usual). But "two times ten to the eight" is common (though less common than "to the eighth").In American English, we'd say "five times ten to the fifth" or "five times ten to the fifth power" for the first one. Similarly, the second would be either "two times ten to the eighth" or "two times ten to the eighth power."
And I agree with that: cardinal numbers for negative exponents.... When the exponents are negative, one rarely (in my experience) uses the ordinal: 4 x 10^(- 3) is said as "four times ten to the minus three" or "four times ten to the negative three." I don't think I have heard anyone use "four times ten to the negative third" in a professional context, though it wouldn't be wrong to say that: it just sounds odd (to me).