Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous chapters we saw how to simplify expressions with multiplied exponential terms
by adding the exponents of terms with like bases
and how to simplify expressions with divided exponential terms
by subtracting the exponents of terms with like bases.
In this lecture we will use these rules
to simplify expressions containing a variety of exponential terms with different bases
which are multiplied and divided.
Our goal will be to write these expressions in their simplest possible form.
We will do this by combining all terms which have the same base.
For example, in this expression there are two terms with a base of two
three terms with a base of x
and two terms with a base of y.
After we have done this, if we are left with any terms with negative exponents
we can change the exponents to positive
by moving those terms from the numerator to the denominator or vice versa.
When combining terms with the same base which are multiplied and divided
we can first add all the exponents in the numerator
and add all the exponents in the denominator
and then subtract the denominator exponent from the numerator exponent.
Or if we like, we can add all the numerator exponents
and subtract all the denominator exponents in one step.
Now let's try simplifying the expression with several different bases
that we saw at the beginning of the lecture.
We'll start by combining all the terms which have a base of two.
We first add an exponent of "one" to the term without an exponent
so that we can subtract exponents.
Then subtracting the exponent of the denominator term from the exponent of the numerator term
we get two to the negative third power in the numerator.
Since this term has a negative exponent
we move the term to the denominator and switch the sign of the exponent to positive.
Since two cubed is equal to eight, we will write it as eight.
Next, we combine all the terms with a base of x.
We first add an exponent of "one" to the term without an exponent
and then, subtracting the exponents of the denominator terms
from the exponent of the numerator term
we get x to the negative second power.
Since this term has a negative exponent
we move it to the denominator and switch the sign of the exponent to positive.
Next, we combine all the terms with a base of y.
Subtracting the exponent of the denominator term from the exponent of the numerator term
we get y to the first power which can just be written as y.
Since there is only a single term with a base of z
we add it to our simplified expression unchanged.
However, since the z term has a negative exponent
we move it to the denominator and switch the sign of the exponent to positive.
The expression is now written in its simplest form.
In our next example, we will simplify another exponential expression
with several different bases.
At first glance, it appears that we only have one term with a base of two.
However, since eight is equal to two cubed
we can temporarily write it that way so that we can combine both terms.
Subtracting the exponent of the term in the denominator
from the exponent of the term in the numerator
we get two to the first power, or just two.
Next, we combine the terms with a base of x.
We first add an exponent of "one" to the term in the numerator without an exponent
and subtract the exponents of the denominator terms.
This gives us x to the zero power.
Since this is the same thing as "one", we can eliminate this term completely.
Next we combine the terms with a base of y.
Subtracting the exponent of the denominator term from the exponent of the numerator term
we get y to the first power, or just y.
Since there is just one term with a base of z
we add it to our simplified expression unchanged.
However, since the z term has a negative exponent
we move it to the numerator and switch the sign of the exponent to positive.
The expression is now written in its simplest form.
In the previous example, we started with an expression written as a fraction
and in the process of simplifying the expression, the denominator disappeared.
This often happens when rearranging terms in an expression
as we sometimes end up moving all the terms to the numerator.
But how can a denominator just disappear?
Here is what actually happens.
When we moved the denominator term to the numerator
we actually took a shortcut, combining several steps.
The first step was to factor out the denominator term from the rest of the expression.
This left an expression with a denominator of "one".
We then used the rule we discovered in the previous lecture
to convert "one" over z to the minus two to z squared.
We then multiplied z squared with the rest of the expression
to combine everything back into one expression.
Since this expression had a denominator of "one" it could be eliminated.
So whenever we move all the terms out of the denominator
we are actually left with a denominator of "one" which we can then eliminate.
Likewise, sometimes when rearranging terms
we end up moving all the terms out of the numerator.
In this case we must add a "one" to the numerator.
But why?
Just as before, when moving the numerator term to the denominator
we actually combine several steps into one action.
The first step is to factor out the numerator term from the rest of the expression.
This leaves an expression with a numerator of "one".
We then use the rule from the previous lecture to convert the term
to "one" over that term with the sign of the exponent switched.
Finally we multiply to combine everything back into one expression.
So whenever we move all the terms out of the numerator
we are left with a numerator of "one".
So far, we have learned rules for multiplying exponential expressions with the same base
and dividing exponential expressions with the same base.
In the next lecture, we will learn what happens
when we raise an exponential expression to a power.