Laws of Exponents
Exponents are also called Powers or Indices
The exponent of a number says how many times to use the number in a multiplication.
In this example: 8^{2} = 8 × 8 = 64
Try it yourself:
So an Exponent saves us writing out lots of multiplies!
Example: a^{7}
a^{7} = a × a × a × a × a × a × a = aaaaaaa
Notice how we wrote the letters together to mean multiply? We will do that a lot here.
Example: x^{6} = xxxxxx
The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x^{2}x^{3} = (xx)(xxx) = xxxxx = x^{5}
Which shows that x^{2}x^{3} = x^{5}, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
The exponent says how many times to use the number in a multiplication.  
A negative exponent means divide, because the opposite of multiplying is dividing  

If you understand those, then you understand exponents!
And all the laws below are based on those ideas.
Laws of Exponents
Here are the Laws (explanations follow):
Law  Example 

x^{1} = x  6^{1} = 6 
x^{0} = 1  7^{0} = 1 
x^{1} = 1/x  4^{1} = 1/4 
x^{m}x^{n} = x^{m+n}  x^{2}x^{3} = x^{2+3} = x^{5} 
x^{m}/x^{n} = x^{mn}  x^{6}/x^{2} = x^{62} = x^{4} 
(x^{m})^{n} = x^{mn}  (x^{2})^{3} = x^{2×3} = x^{6} 
(xy)^{n} = x^{n}y^{n}  (xy)^{3} = x^{3}y^{3} 
(x/y)^{n} = x^{n}/y^{n}  (x/y)^{2} = x^{2} / y^{2} 
x^{n} = 1/x^{n}  x^{3} = 1/x^{3} 
And the law about Fractional Exponents:  
x^{m/n} =
n√x^{m}
= (n√x )^{m} 
x^{2/3} =
3√x^{2}
= (3√x )^{2} 
Laws Explained
The first three laws above (x^{1} = x, x^{0} = 1 and x^{1} = 1/x) are just part of the natural sequence of exponents. Have a look at this:
Example: Powers of 5  

.. etc..  
5^{2}  1 × 5 × 5  25  
5^{1}  1 × 5  5  
5^{0}  1  1  
5^{1}  1 ÷ 5  0.2  
5^{2}  1 ÷ 5 ÷ 5  0.04  
.. etc.. 
Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).
The law that x^{m}x^{n} = x^{m+n}
With x^{m}x^{n}, how many times do we end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
Example: x^{2}x^{3} = (xx)(xxx) = xxxxx = x^{5}
So, x^{2}x^{3} = x^{(2+3)} = x^{5}
The law that x^{m}/x^{n} = x^{mn}
Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we are dividing), for a total of "mn" times.
Example: x^{4}/x^{2} = (xxxx) / (xx) = xx = x^{2}
So, x^{4}/x^{2} = x^{(42)} = x^{2}
(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.)
This law can also show you why x^{0}=1 :
Example: x^{2}/x^{2} = x^{22} = x^{0} =1
The law that (x^{m})^{n} = x^{mn}
First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.
Example: (x^{3})^{4} = (xxx)^{4} = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x^{12}
So (x^{3})^{4} = x^{3×4} = x^{12}
The law that (xy)^{n} = x^{n}y^{n}
To show how this one works, just think of rearranging all the "x"s and "y"s as in this example:
Example: (xy)^{3} = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x^{3}y^{3}
The law that (x/y)^{n} = x^{n}/y^{n}
Similar to the previous example, just rearrange the "x"s and "y"s
Example: (x/y)^{3} = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x^{3}/y^{3}
The law that x^{m/n} = n√x^{m} = (n√x )^{m}
OK, this one is a little more complicated!
I suggest you read Fractional Exponents first, so this makes more sense.
Anyway, the important idea is that:
x^{1/n} = The nth Root of x
And so a fractional exponent like 4^{3/2} is really saying to do a cube (3) and a square root (1/2), in any order.
Just remember from fractions that m/n = m × (1/n):
Example: x^{(mn)} = x^{(m × 1n)} = (x^{m})^{1/n} = n√x^{m}
The order does not matter, so it also works for m/n = (1/n) × m:
Example: x^{(mn)} = x^{(1n × m)} = (x^{1/n})^{m} = (n√x )^{m}
Exponents of Exponents ...
What about this example?
4^{32}
We do the exponent at the top first, so we calculate it this way:
Start with:  4^{32}  
3^{2} = 3×3:  4^{9}  
4^{9} = 4×4×4×4×4×4×4×4×4:  262144 
And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when you understand the
three ideas near the top of this page:
 The exponent says how many times to use the number in a multiplication
 A negative exponent means divide
 A fractional exponent like 1/n means to take the nth root: x^{(1n)} = n√x
Oh, One More Thing ... What if x = 0?
Positive Exponent (n>0)  0^{n} = 0  
Negative Exponent (n<0)  0^{n} is undefined (because dividing by 0 is undefined)  
Exponent = 0  0^{0} ... ummm ... see below! 
The Strange Case of 0^{0}
There are different arguments for the correct value of 0^{0}
0^{0} could be 1, or possibly 0, so some people say it is really "indeterminate":
x^{0} = 1, so ...  0^{0} = 1  
0^{n} = 0, so ...  0^{0} = 0  
When in doubt ...  0^{0} = "indeterminate" 