Use with ordinal numbers (first, tenth, hundredth) is somewhat more common and formal for positive integers.

For negative, zero, or fractional powers, the ordinal form (minus tenth, zeroth, halfth) is rare or non-existent without the word "power".

The use with cardinal numbers (one, ten, one hundred, zero, minus ten) is common in Mathematics and Computing and is almost always used for numbers other than positive integers.

Formal usage with ordinals normally includes the word "power" for clarity, and the alternative form to the power of is preferred for powers that are not positive integers.