What would be the correct way of writing 10⁻³⁶ second into words?
Note, "negative" as used in this answer is interchangeable with "minus". Thanks to a number of commenters pointing this out, NGram Viewer and Google search both show "minus" as being more frequent than "negative". "Negative" may be more common in formal use.
Exponentiation such as you've shown is usually said, in your example either:
Ten raised to the power of negative 36
or
Ten to the power of negative 36
or
Negative 36th power of 10 (less common)
or
Ten to the negative 36th power
or more casually as:
Ten to the negative 36th
Notice that in some cases we usually use the cardinal number (36) whereas in other cases we use the ordinal number (36th) when referring to the exponent. Although what I've listed are the more common ways, the cardinal and ordinal forms aren't absolutely universal. For example, you may hear:
10 to the negative 36
However, in my opinion this is much less common.
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The exponent is usually shown as a superscript to the right of the base. In that case, b^n is called "b raised to the n-th power", "b raised to the power of n", or "the n-th power of b".
Wikipedia: Exponentiation
Notice how it's said in this Khan Academy video, it's said in the shorter way I described, ie., he says ten to the twenty-third.
Specifically in your case of seconds, it would be said:
Ten to the negative 36th seconds
or
Ten raised to the power of negative 36 seconds
or
Ten to the power of negative 36 seconds
or
36th power of 10 seconds (somewhat less common in my opinion, I don't recommend this)
Finally, about the word "undecillion" or "undecillionth", I would advise against using that unless comprehension is not important. Firstly, it's an uncommon word. Secondly, as shown by the dictionary definition below, it can be two different numbers, I assume as a result of the differences between the short and long scales which (historically) separated Europe and the US.
undecillion
a cardinal number represented in the U.S. by 1 followed by 36 zeros, and in Great Britain by 1 followed by 66 zeros.
Random House Unabridged Dictionary