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

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!

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

Dividing

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: