# Variables with Exponents

## 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 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):

(Remember: a variable without an exponent really has an exponent of 1, example: y is y1)

## 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 = 1x x-2 = 1x2 x-3 = 1x3 etc...

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: