Laws of Exponents
Exponents are also called Powers or Indices

The exponent of a number says how many times to multiply the number.
In this example: 8^{2} = 8 × 8 = 64
 In words: 8^{2} could be called "8 to the second power", "8 to the power 2" or
simply "8 squared"

All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents"), all come from three ideas:

The exponent of a number says to multiply the number by itself so many times 



The opposite of multiplying is dividing, so a negative exponent means divide 




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^{4}/x^{2} = x^{42} = x^{2} 
(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} 




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:
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.. 

You will see that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 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 will you 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 will you end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because you are dividing), for a total of "mn" times.
Example: x^{42} = x^{4}/x^{2} = (xxxx) / (xx) = xx = 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 x "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" 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
To understand this, just remember from fractions that n/m = n × (1/m):
Example:
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) 
0^{n} = 0 
Negative Exponent (n<0) 
Undefined! (Because dividing by 0) 
Exponent = 0 
Ummm ... see below! 
The Strange Case of 0^{0}
There are two different arguments for the correct value. 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" 
