The law that xmxn = xm+n
With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
x2x3 = (xx) × (xxx) = xxxxx = x5
So, x2x3 = x(2+3) = x5
The law that xm/xn = xm-n
Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because you are dividing), for a total of "m-n" times.
x4-2 = x4/x2 = (xxxx) / (xx) = xx = x2
(Remember that x/x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)
The law that (xm)n = xmn
First you multiply x "m" times. Then you have to do that "n" times, for a total of m×n times.
Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12
So (x3)4 = x3×4 = x12
The law that (xy)n = xnyn
To show how this one works, just think of re-arranging all the "x"s and "y" as in this example:
(xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3
The law that (x/y)n = xn/yn
Similar to the previous example, just re-arrange the "x"s and "y"s
(x/y)3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x3/y3
The law that
To understand this, just remember from fractions that n/m = n × (1/m):
And That Is It
If you find it hard to remember all these rules, then remember this:
you can always work them out if you understand the three ideas at the top of this page.
Oh, One More Thing ...
What if x= 0?
Positive Exponent (n>0) 0n = 0
Negative Exponent (n<0) Undefined! (Because dividing by 0)
Exponent = 0 Ummm ... see below!
The Strange Case of 00
There are two different arguments for the correct value. 00 could be 1, or possibly 0, so some people say it is really "indeterminate":
x0 = 1, so ... 00 = 1
0n = 0, so ... 00 = 0
When in doubt ... 00 = "indeterminate"