How do you say "powers of ten"?

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".

I express 3^4 as “three to the fourth power”

You can say “base to the nth power” or “base to the power of n”

It’s important to have the whole sentence to determine if it makes mathematical sense.

The term power refers to the exponent, not to the base.

10 to the power 2 is 100.

However powers of 10 are the products obtained from raising 10 by various exponents. So again, power does not refer to the base.

A common expression for power(s) of 10 in regular speech is order(s) of magnitude.

From Wikipedia:

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.