Variables with Exponents

How to Multiply and Carve up 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 all the same. It is usually a letter similar ten or y.

An exponent (such as the 2 in xtwo ) says how many times to use the variable in a multiplication.

Example: y2 = yy

(yy ways y multiplied by y, because in Algebra putting two letters next to each other ways to multiply them)

Likewise z3 = zzz and x5 = xxxxx

Exponents of 1 and 0

Exponent of one

When the exponent is 1, we only have the variable itself (example 101 = ten)

We commonly don't write the "one", but information technology sometimes helps to remember that ten is likewise xane

Exponent of 0

When the exponent is 0, we are not multiplying by anything and the answer is just "1"
(case y0 = 1)

Multiplying Variables with Exponents

So, how practice we multiply this:

(yii)(ythree)

We know that ytwo = yy, and y3 = yyy so allow us write out all the multiplies:

y2 yiii = yy yyy

That is 5 "y"s multiplied together, so the new exponent must be 5:

y2 y3 = yfive

But why count the "y"s when the exponents already tell usa how many?

The exponents tell us at that place are ii "y"s multiplied by 3 "y"southward for a full of five "y"s:

ytwo y3 = y2+3 = y5

So, the simplest method is to just add the exponents!

(Note: this is ane of the Laws of Exponents)

Mixed Variables

When we accept a mix of variables, but add upwardly 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

At that place will often exist constants (numbers similar three, 2.9, ½ etc) mixed in as well.

Never fear! Merely multiply the constants separately and put the event in the answer:

(Note: "·" means multiply, which we employ when the "×" might be confused with the alphabetic character "ten")

Hither is a more than complicated case with constants and exponents:

Negative Exponents

Negative Exponents Mean Dividing!

x-ane = 1 x x-2 = 1 10two x-3 = 1 xthree etc...

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

Dividing

And so, how practice nosotros do this? y3 y2

Let'south write out all the multiplies: yyy yy

Now remove any matching "y"southward that are
both top and lesser (because y y = 1)

And we are left with: y

So three "y"s above the line become reduced past 2 "y"s below the line, leaving but 1 "y" :

ythree y2 = yyy yy = y3-2 = yi = y

OR, we could take done it like this:

y3 yii = y3y-two = y3-2 = y1 = y

So ... only 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 "cantankerous out" the variables that are both top and bottom:

x3 y z2 x ytwo zii = xxx y zz x yy zz = tenxx y zz x yy zz = xx y = tenii y

But one time again, why count the variables, when the exponents tell yous how many?

In one case yous get confident yous tin can exercise the whole thing quite quickly "in place" like this: