z. B. 10 to the power of 10

Colloquially also shortened to "x to the y".

Colloquially also shortened to "x to the y".

Surely "x to the yth"?

yup sorry, I was dithering about how to deal with "nd" "rd" and "th" and then got sidetracked.

You don't use the ordinals (nd, rd, th) when shortening to "x to the y" in British English, because you are justing omitting "power of"

i.e. 10^4 is "10 to the four", not "10 to the fourth", because it's actually "10 to the power of 4".

In the US, I have heard almost exclusively "to the N-th power" or just "to the N-th". 10 hoch 10 heißt "ten to the tenth (power)".