When you have powers of 10, e.g. 102, the base is 10, so when the exponent is 2 you should not say power of 2. I believe "power of" refer to the base not to the exponent.
While "ten to the power of two" is correct (and the "power" does indeed refer to the "two" in this construction), it's also possible and very common to drop the "power of", giving "ten to the two". When reading out vacuum pressures for example, "ten to the power of minus six" would never be heard from a native speaking physicist; we'd just say "ten to the minus six". This is equally true in longer constructions like "three point five times ten to the minus seven".
A common expression for power(s) of 10 in regular speech is order(s) of magnitude.
An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system. It is equal to the logarithm (base 10) rounded to a whole number. For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 10^3.
Surprisingly, this is explained fairly well on Wikipedia.
I believe "power of" refer to the base not to the exponent
Nope. The spoken forms of 102 are:
- 10 raised to the second power, or
- 10 raised to the power of two, or
- 10 to the power of two, or
- 10 to the two, or simply
- 10 squared
Since the original formulation base raised to the nth power means multiply 1 by base n times, the word power does indeed refer to the exponent.
"Powers of 10" does definitely refer to power expressions with 10 as a base rather than as an exponent. I don't have any sourced explanation (which makes this a terrible answer), but I imagine it's because of the similarity between the two phrases
- 10 raised to the second power
- the second power of 10
The expression a power of 10 typically means the number you get when you raise 10 to a power (exponent, in other words) which itself is a number. I know it's a little bit confusing since you refer to the result of raising a number to a power also as a power, but that's just how people say it. Thus, you can say that the following is a list of powers of 10, that is, a list of the numbers you get when you raise 10 to a particular power such as 1, 2, 3, etc:
101 = 10
102 = 100
103 = 1000
Given the fact that the numbers 102 and 100 are equivalent, they both can be referred to as a power of ten. More specifically, it's ten raised to the second power or more compactly ten to the second power. Likewise, 108 would be pronounced ten to the eighth power or ten raised to the eighth power.
Usually, for powers that are greater than 3, you can drop the word "power". For example, instead of saying ten to the eighth power, you can just say ten to the eighth.