Variables with Exponents

How to Multiply and Divide them

What is a Variable with an Exponent?

variable x with exponent 2

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 = yy yyy

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, 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 = 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?y3y2
Let's write out all the multiplies:yyyyy

Now remove any matching "y"s that are
both top and bottom (because yy = 1)

And 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" :

y3y2 = yyyyy = y3-2 = y1 = y

OR, we could have done it like this:

y3y2 = 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 z2x y2 z2=xxx y zzx yy zz=xxx y zzx yy zz=xxy=x2y

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: