Welcome to a lesson on properties of exponents.
In this lesson,
we'll review the multiplication property of exponents,
as well as raising a power to a power.
But let's begin by reviewing
some basic exponent properties given here.
First we have n raise to the 1st power.
Remember the exponent tells us
how many factors we have of the base,
so here we have one factor of n which equals n.
So any real number raised to the 1st power,
is always equal to the base.
Next we have one raise to the power of n.
Whenever we have a base of one raised to any exponent,
it's always equal to one.
Next we have n to the zero,
where the base n is not zero.
Whenever we have a non-zero real number
raised to the zero power, it's always equal to one.
Now we'll try to justify this using a pattern
in just a moment.
Finally we have zero raise to the power of n,
where the exponent of n is not zero.
Zero raised to any non-zero exponent,
is always equal to zero.
But let's try to justify why n to the zero must equal one.
Notice here we have powers of three.
We start with three to the 4th,
and in each row the exponent decreases by one.
Well three to the 4th is equal to 81,
and notice how if we take 81 and divide by three,
which is equal to 27,
we get three to the 3rd.
Continuing this pattern,
27 divided by three is equal to nine,
which also equals three squared.
Nine divided by three is equal to three
which equals three to the 1st,
and therefore three divided by three which equals one,
must equal three to the zero.
So here's some justification
as to why n raised to the power of zero equals one,
as long as the base doesn't equal zero.
Of course this pattern does continue,
one divided by three equals 1/3,
which equals three to the power of negative one.
So this pattern continues to work
even for negative exponents.
Now let's take a look at example one.
Here we have five x to zero.
We need to be careful here
because the exponent of zero is only attached to the x.
This means 5 times x to the zero,
where x to zero equals one because we're told,
the base x doesn't equal zero.
So this is equal to five times one, which equals five.
Next we have the quantity two x plus one,
raised to the power of zero.
We're told x doesn't equal negative 1/2,
which would make our base zero.
So we have a non-zero base raised to the zero power,
and therefore this equals one.
Here we're given that a, b, and c,
the three bases are not zero.
So a to the zero equals one,
b to the zero equals one, and c to the zero equals one,
and therefore this simplifies to three.
Now let's review the multiplication property of exponents.
Let's take a look at here in red.
If we're multiplying and the bases are the same,
we add the exponents.
And let's take a moment and justify this.
If we have x to the 2nd times x to the 3rd,
we know x to the 2nd is equal to two factors of x.
X to the 3rd is equal to three factors of x,
and therefore this product is equal to x to the 5th.
But instead of writing out each factor
notice how we could just add the exponents,
giving us x to the 5th.
So for n to the 3rd times n to the 9th,
because we're multiplying and the bases are the same,
we add the exponents.
So this is equal to n raised to the power of three plus nine
which equals n to the 12th .
Next we have b to the 5th times b to the 4th times b,
b is the same as b to the 1st.
So again we're multiplying and the bases are the same,
so we add the exponents.
So this is b to the power of five plus four plus one,
what is equal to b to the 10th.
Next we have five x squared y to the 5th
times seven x y to the 9th.
Let's include an exponent of one on the x,
and now when we multiply,
we'll first multiply the coefficients.
Five times seven equals 35,
and we have x to the 2nd times x to the 1st,
that's x to the power of two plus one.
And then have y to the 5th times y to the 9th,
so we have y to the power of five plus nine,
so this simplifies to 35,
x to the 3rd, y to the 14th.
Now as for the exponent property,
when raising a power to a power.
If we have a raise to the power of m,
raise to the power of n,
well we have a power raise to power,
we multiply the exponents.
This is equal to a raise the power of m times n,
and let's quickly justify this property as well.
Let's say we had x to the 2nd raise to the 3rd,
we know something is raised to the 3rd power,
we have three factors of the base shown here.
So notice how here we have total of six factors of x
and therefore this simplifies to x to the 6th.
But a much shorter way to determine x to the 6th,
would be to simply multiply the exponents.
Two times three, gives us an exponent of six,
which is the same as two plus two plus two.
So going back to example three,
here we have x to the 3rd raised to the 9th.
So because we have powers raise to powers,
we multiply the exponents.
This is equal to x raised to the power of three times nine,
which is equal to x to the the 27th.
Next we have five b times five b to the 5th
raised to the 8th.
We first simplify the powers to powers,
so we'd have five b to the 2nd,
times b raised to the power of five times eight, that's 40.
And now here we're multiplying and the bases are the same,
so we add the exponents.
So we have five b raised to the power of two plus 40
which is 42.
And next we have a rule for raising a product to a power.
Here we have a, b, raised to the power of n,
is equal to a to the nth, times b to the nth.
This is just an extension of raising a power to a power
which we did above.
Notice inside the parentheses,
we have a to the 1st times b to the 1st,
raised to the nth.
So here we are multiplying the exponents
because we have powers to powers.
So we're going to go ahead
and just reuse the same property we did above
for the following examples.
So here we have five x to the 2nd,
which is the same as five to the 1st,
x to the 1st of the 2nd.
So because we have powers to powers
we multiply the exponents,
so we'll have five raised to power of one
times two of five squared,
times x raise to power of one times two x squared,
and five squared equals 25,
so this simplifies to 25x squared.
Next we have x to the 3rd y squared raise to the 9th.
Again we have powers to powers,
so we multiply the exponents.
Here we'll have x raise to the power of three times nine,
and y raise to the power of two times nine.
So this simplifies to x raise to the 27th
and y raise to the 18th.
Next we have negative eight a, b to the 5th squared.
We need to be careful because of this negative eight here.
Let's write this negative eight in parenthesis
and then raise it to the 1st power,
and then a would also an exponent of one.
So again because we have powers to powers,
we multiply the exponents
so we'd have negative eight in parentheses,
raise to the power of one times two that's two,
and then we'd have a, raise to the power of one times two,
which is two,
and then we'd have b raise to the power of five times two
which is 10,
and negative eight squared would be positive 64.
So we have positive 64a squared, b to the 10th.
Now let's take a look at our last two examples.
Here we have five times negative two w to the 7th cubed.
Again because we have a negative factor here,
we need to be careful.
Let's put this negative two in parentheses
and this would be raised to the first.
So because we have powers raised to powers
we multiply the exponents, so we'll have five times,
we'd have negative two raise to power of one times three,
or three, and then w,
raise to power of seven times three which is 21.
Negative two cubed is negative eight,
so we have five times negative eight,
w to the 21st,
so we have negative 40w to the 21st.
And in our last example,
notice how we have five n to the 4th
times negative three n cubed squared.
So here we have a negative three
which is negative three in parenthesis raised to the 1st,
and cubed squared so we have powers to powers.
So our first step we have five n to the 4th
times negative three to the power of one times two,
just two,
and then we have n raised to the power three times two
which is six,
so simplifying we have five n to the 4th,
times negative three squared that'd be nine,
so times nine n to the 6th,
and five times nine is 45,
n to the 4th times n to the 6th
is equal to n to the power of four plus six
which would be 10.
Okay that's going to do it for this lesson.
I hope you found this helpful.