I think maybe I'm doing this just so I can have something to point people at when they are wrong on the internet... but still, some of you might find it useful to have a different perspective when you're messing around with powers. It's not rigorous, because I've forgotten how to write out proper deductions.

1) 2^2 = 2*2 = 4

2) 2^3 = 2*2*2 = 8

3) 2^4 = 2*2*2*2 = 16

4) 2^2 * 2 = 2*2*2 = 8

5) 2^3 / 2 = 2*2*2 / 2 = 4

sub (1) into (5)

6) 2^3 / 2 = 2^2

infer

7) 2^3 / 2 = 2^(3-1)

8) 2^x / 2 = 2^(x-1)

sub 2 into x in (8)

9) 2^2 / 2 = 2^1

10) 2^2 / 2 = 2*2 / 2 = 2

10) 2^1 = 2

sub 1 into x in (8)

11) 2^1 / 2 = 2^0

12) 2^1 / 2 = 2 / 2 = 1

13) 2^0 = 1

sub 0 into x in (8)

14) 2^0 / 2 = 2^-1

15) 2^0 / 2 = 1 / 2

16) 2^-1 = 1 / 2

thus we see the reason why reciprocal (1/n) is also called ^-1

and now fractions:

17) 2^2 = 2 * 2 = 2^1 * 2^1 = 2^2 * 2^0

18) 2^(2-x) * 2^x = 2^2

19) 2^(n-x) * 2^x = 2^n

sub 1 for n in (19)

20) 2^(1-x) * 2^x = 2^1

sub 1/2 for x in (20)

21) 2^(1/2) * 2^(1/2) = 2^1

which explains why the square root is called ^(1/2)

for any positive value (a), (a^(1/2))^2 = (a)

to specify the cube root x of a (x*x*x = a) we need to divide the power into three (x = a^(1/3))

to specify the change in pitch of a semi tone, we know that the pitch doubles per octave, and that there are 12 notes in every octave, that's pitchchange = 2^(1/12)

If you have a game and you want something to increase in power, then you specify how much you want it to be finally at, and use power of 1 over the number of steps to get there to do it smoothly.

price of base thing (b) goes to price of super expensive (x), over four steps

price (step) = b * (x/b) ^ ((1/4) * step)

price (0) = b * (x/b) ^ ((1/4)*0) = b * (x/b) ^ 0 = b * 1

price (4) = b * (x/b) ^ ((1/4)*4) = b * (x/b) ^ 1 = b * (x/b) = x

price (2) = b * (x/b) ^ ((1/4)*2) = b * (x/b) ^ (1/2) = b * sqrt(x/b)