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 x2) says how many times
Example: y2 = yy
(yy means y multiplied by y, because in Algebra putting two letters next to each other means to multiply them)
Likewise z3 = zzz and x5 = xxxxx
Exponents of 1 and 0
Exponent of 1
When the exponent is 1, we just have the variable itself (example x1 = x)
We usually don't write the "1", but it sometimes helps to remember that x is also x1
Exponent of 0
When the exponent is 0, we are not multiplying by anything and the answer is just "1"
(example y0 = 1)
Multiplying Variables with Exponents
So, how do we multiply this:
We know that y2 = yy, and y3 = yyy so let us write out all the multiplies:
y2 y3 = yyyyy
That is 5 "y"s multiplied together, so the new exponent must be 5:
y2 y3 = y5
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:
y2 y3 = y2+3 = y5
So, the simplest method is to just add the exponents!
(Note: this is one of the Laws of Exponents)
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 y1)
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 Mean Dividing!
|x-1 = 1x||x-2 = 1x2||x-3 = 1x3||etc...|
Get familiar with this idea, it is very important and useful!
|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:
So 3 "y"s above the line get reduced by 2 "y"s below the line, leaving only 1 "y" :
|y3||=||yyy||= y3-2 = y1 = y|
OR, we could have done it like this:
|y3||= y3y-2 = y3-2 = y1 = y|
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 z2/z2 = 1)
To see what is going on, write down all the multiplies, then "cross out" the variables that are both top and bottom:
|x3 y z2||=||xxx y zz||=||=||xx||=||x2|
|x y2 z2||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: