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. An exponent (such as the 2 in x^{2}) says how many times 
Example: y^{2} = yy
(yy means y multiplied 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
When the exponent is 1, we 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
When the exponent is 0, we are not multiplying by anything and the answer is just "1"
(example y^{0} = 1)
Multiplying Variables with Exponents
So, how do we 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 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
When we have a mix of variables, just add up the exponents for each, like this (press play):
(Remember: a variable without an exponent really has an exponent of 1, example: y is y^{1})
With Constants
There will often be constants (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: "·" means 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} = \frac{1}{x}  x^{2} = \frac{1}{x^{2}}  x^{3} = \frac{1}{x^{3}}  etc... 
Get familiar with this idea, it is very important and useful!
Dividing
So, how do we do this? 


Let's write out all the multiplies: 


Now 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" :
y^{3}  =  yyy  = y^{32} = y^{1} = y 
y^{2}  yy 
OR, we 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 we 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)
To see what is going on, write down all the multiplies, then "cross out" the variables that are both top and bottom:
x^{3} y ^{}z^{2}  =  xxx y zz  =  =  xx  =  x^{2}  
x y^{2} z^{2}  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: