# 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

^{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**so let us write out all the multiplies:

^{3}= yyyy^{2} y^{3} = yy yyy

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, which we use when the "×" might be confused with 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

Now remove any matching "y"s that are

both top and bottom (because \frac{y}{y} = 1)

So 3 "y"s above the line get reduced by 2 "y"s below the line, leaving only 1 "y" :

\frac{y^{3}}{y^{2}} = \frac{yyy}{yy} = y^{3-2} = y^{1} = y

**OR, we could have done it like this:**

\frac{y^{3}}{y^{2}} = y^{3}y^{-2} = y^{3-2} = y^{1} = 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 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:

\frac{x^{3} y z^{2}}{x y^{2} z^{2}}=\frac{xxx y zz}{x yy zz}=\frac{xxx y zz}{x yy zz}=\frac{xx}{y}=\frac{x^{2}}{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: