Hello. I’m Professor Von Schmohawk and welcome to Why U.
We have seen that any number "a" raised to the nth power, where n is a positive integer,
is equivalent to n a's multiplied together.
We also saw that to multiply exponential expressions with the same base, we add their exponents.
In this lecture, we will show how to use these rules
to write exponential expressions in their simplest possible form.
A group of numbers and/or letters combined using mathematical operations
is called a "mathematical expression".
The individual elements which are added, subtracted, multiplied, or divided to create the expressions
are sometimes referred to as "terms".
There are several ways to write products of terms.
As we saw earlier, when multiplying numbers,
we can either use the standard multiplication symbol or a dot.
In our lectures, we will typically use a dot to indicate multiplication
so that it will not be confused with the letter "x".
Often, instead of using specific numbers, we use letters to indicate any possible number.
These letters are referred to as "variables",
since unlike numbers which always have a specific value,
their value can vary to represent any possible value.
When using variables,
we have one additional option for indicating that two variables are multiplied.
We can write the variables next to each other with nothing in between.
We can also use this notation when multiplying a variable and a number.
We can also do this when these terms have exponents.
When different variables are used in an expression,
it is assumed that they can each have different values.
For example, x squared times y cubed cannot be combined by adding their exponents
since x and y may represent different bases.
On the other hand, when the same variable is used more than once in an expression,
it is assumed that whatever value it represents is the same throughout the expression.
x squared times x cubed can be combined by adding their exponents
since both x’s represent the same base.
Now let’s look at some examples of exponential expressions
and see if we can write them in a simpler form.
In our first example we have two terms,
x to the fifth power and x cubed, multiplied together.
Since these terms have the same base, x,
we can add their exponents,
and the result is x to the eighth power.
It is important to remember that we can only add exponents when the terms are multiplied.
If these two terms were added instead of multiplied,
there would be no way to simplify this expression any further.
For our next example, let’s simplify x cubed times x.
Since x is the same as x to the first power,
let’s write it that way so that we can add the exponents.
Now adding the exponents
gives us x to the fourth power.
Now let’s simplify an expression with three terms,
"a" to the fourth times "a" times "a" cubed.
Remember that it doesn’t matter what letter is used for a variable.
There is no difference between using an "a" or an "x" or any other letter.
We start by writing the "a" without an exponent as "a" to the first power
so we can add the exponents.
And adding the exponents, we get "a" to the eighth power.
Let’s try an example with two different variables.
Remember that we can only add exponents of terms whose bases are the same.
Therefore, we can add the exponents for the x’s and the exponents for the y’s
but we cannot combine the x and y terms.
The result is x to the ninth times y to the fifth.
In the next example, the terms in the expression have bases which are numbers instead of variables.
Let’s simplify two cubed times two to the fourth.
Adding the exponents, we get two to the seventh power,
which is the same as seven twos multiplied together,
or 128.
In our last example, we will simplify a complicated-looking exponential expression
with several variables and numbers.
A good place to start is to group like bases together.
Let’s move all the twos to the front,
followed by the threes,
then the x’s, y’s, and z’s.
Now add an exponent of one to any term that has no exponent,
and combine exponents of terms which have the same base.
This gives us two to the fifth,
times three cubed,
times x to the fourth,
times y to the seventh,
times z squared.
If we like, the numeric terms may be kept separate,
or we can simplify this a bit further by combining them into a single number.
Two to the fifth is thirty-two,
and three cubed is twenty-seven,
and thirty-two times twenty-seven is 864.
So far we have simplified exponential expressions where the terms in the expression are multiplied.
In the next lecture we will demonstrate examples where the terms are divided.