In standard form, if the power of 10 you need is just 10 or 10^1, is it usual practice to include the power '1' in your answer?

I think that 10 would be acceptable.

No.
One is normally omitted.

Only a nitpicking teacher, instructor, or professor worries about the exponent 1, which is automatically implied if absent. In a professional scientist's notebook, unambiguous accuracy is far more important than including unnecessary minutia.

If you want to use the very strict (formal) scientific notation, then yes.
And it also applies to units (10 to the power zero)

7.5
becomes 7.5 * 10^0
or 7.5E0 (on some calculators or spreadsheets).

63.2
becomes 6.32*10^1
or 6.32E1

and so on.

The only exception is 0
0
You CAN write it as 0 * 10^0
or 0E0
but 0 by itself is always zero.