Variables with Exponents

How to Multiply and Divide them

What is a Variable with an Exponent?

variable with 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 x2) says how many times
to use the variable in a multiplication.

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:

(y2)(y3)

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)

Mixed Variables

When we have a mix of variables, just add up the exponents for each, like this (press play):

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 = 1   x-2 = 1   x-3 = 1
x x2 x3

Get familiar with this idea, it is very important and useful!

Dividing

So, how do we do this?  
y3
y2
     
Let's write out all the multiplies:  
yyy
yy
     
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" :

y3 = yyy = y3-2 = y1 = y
y2 yy

OR, we could have done it like this:

y3 = y3y-2 = y3-2 = y1 = y
y2

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 = xxx y zz = xx = x2
x y2 z2 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: