Variables with Exponents
How to Multiply and Divide them
What is a Variable with an Exponent?

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.
An exponent (such as the 2 in x^{2}) says how many times to use the variable in a multiplication. 
Example: y^{2} = yy
(yy means y multipled by y, because in Algebra putting two letters next to each other means to multiply them)
Likewise z^{3} = zzz and x^{5} = xxxxx
Exponents of 1 and 0
Exponent of 1
If the exponent is 1, then you just have the variable itself (example x^{1} = x)
We usually don't write the "1", but it sometimes helps to remember that x is also x^{1}
Exponent of 0
If the exponent is 0, then you are not multiplying by anything and the answer is just "1" (example y^{0} = 1)
Multiplying Variables with Exponents
So, how do you multiply this:
(y^{2})(y^{3})
We know that y^{2} = yy, and y^{3} = yyy so let us write out all the multiplies:
y^{2} y^{3} = yyyyy
That is 5 "y"s multiplied together, so the new exponent must be 5:
y^{2} y^{3} = y^{5}
But why count the "y"s when the exponents already tell us how many?
The exponents tell us that there are two "y"s multiplied by 3 "y"s for a total of 5 "y"s:
y^{2} y^{3} = y^{2+3} = y^{5}
So, the simplest method is to just add the exponents! (Note: this is one of the Laws of Exponents)
Mixed Variables
If you have a mix of variables, just add up the exponents for each, like this (press play):
With Constants
There will often be contants (numbers like 3, 2.9, ½ etc) mixed in as well.
Never fear! Just multiply the constants separately and put the result in the answer:
(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")
Here is a more complicated example with constants and exponents:
Negative Exponents
Negative Exponents Mean Dividing!
x^{1 }= 
1^{} 

x^{2 }= 
1 

x^{3 }= 
1 



x 
x^{2} 
x^{3} 
Get familiar with this idea, it is very important and useful!
Dividing
So, how do you do this? 





If we write out all the multiplies we get: 





We can remove any matching "y"s that are both top and bottom (because y/y = 1), so we are left with: 

y 
So 3 "y"s above the line get reduced by 2 "y"s below the line, leaving only 1 "y" like this:
y^{3} 
= 
yyy 
= y^{32} = y^{1} = y 


y^{2} 
yy 
OR, you could have done it like this:
y^{3} 
= y^{3}y^{2} = y^{32} = y^{1} = y 

y^{2} 
So ... just subtract the exponents of the variables you are dividing by!
Here is a bigger demonstration, involving several variables:
The "z"s got completely cancelled out! (Which makes sense, because z^{2}/z^{2} = 1)
You can see what is going on if you write down all the multiplies, then "cross out" the variables that are both top and bottom:
x^{3} y ^{}z^{2} 
= 
xxx y zz 
= 
xxx y zz 
= 
xx 
= 
x^{2} 





x y^{2} z^{2} 
x yy zz 
x yy zz 
y 
y 
But once again, why count the variables, when the exponents tell you how many?
Once you get confident you can do the whole thing quite quickly "in place" like this:
