Today I'm going to show you everything about how to work with exponents.
So that means we're going to cover all of the basic exponent rules.
We're going to talk about where exponents come in order of operations, and how to deal
with negative exponents and exponents that are fractions.
So if keeping track of all of the exponent rules confuses you, you're definitely in the
right place.
I want to let you know that there are lots of timestamps in the description of this video,
so if you're looking for something in particular about exponents, feel free to skip around
and go straight to the section you want to.
So the first thing we're going to talk about is "What are exponents?".
There's no difference between exponents and powers.
Sometimes people think that they're a different thing, but they mean exactly the same thing.
So we could just as easily be asking "What are powers?".
So exponents are also known as powers.
When we talk about exponents or powers in algebra, or math in general, what we're talking
about is a shorthand way of expressing repeated multiplication of one thing by itself.
So what does that mean?
Well let's say for example that we have this expression, 4 to the 3.
The number on the bottom, 4, is called the base, and this little number on the top right
is called the exponent.
And when we have a base raised to an exponent, what this tells us is that we want to multiply
the base by itself however many times the exponent says.
So because this is 4 to the 3, it means multiply 4, the base, by itself 3 times, because the
exponent is 3.
So this is the same thing as 4 times 4 times 4 because we're multiplying 4 by itself 3
times.
So the reason that we use exponents in math is because if we want this value, 4 times
4 times 4, it's a lot easier to write it as 4^3 instead of writing out this whole thing
4 times 4 times 4 every time.
You'll also see a more generalized example, a to the n, which just means multiply a by
itself n times, in the same way that 4 to the 3 means multiply 4 by itself 3 times.
So whatever number you have for n, or the exponent, that's how many times you multiply
the base by itself.
So this would be a times a times a, dot dot dot, times a and this would be done n times,
because n the exponent here, is how many times we want to multiply a by itself.
Now keep in mind that there are a bunch of different ways to read an exponent.
So for example if we go back to this first problem 4^3, we could read this as "4 to the
3", "4 to the 3rd power", "4 to the 3rd".
So all of those ways of reading this exponential expression are correct.
But usually the simplest way to read this is "4 to the 3rd" or this expression here
as "a to the n".
Now one other thing about how to read exponents.
You can always just say "the base to the exponent" like we did "4 to the 3rd", but there are
a couple of exponents that also have special names.
So for example if you have this, you could definitely read that as "x to the 2".
But because the exponent of 2 is so common, we give it a special name, and we can also
read this as "x squared".
So whenever you see an exponent of 2, you can just say "squared" instead.
So you can read this correctly as "x to the 2" or "x to the 2nd", but more conveniently
you can also read it as "x squared".
Same thing with x to the 3 or x to the 3rd.
The exponent of 3 has a special name called "cubed" so we can also read this as "x cubed".
And you can remember that because when you think of a square, a square has two sides.
If we draw a square we have one side and then the other side, which we multiply together
to get area.
So that's why we say "x squared", because there's two sides and we have an exponent
of 2.
If you have an exponent of 3, we say "cubed" because if we draw a cube like this, we have
1, 2, 3 dimensions, right?
Length, width, and height, that we multiply together to get volume, and that's why we
say "x cubed" when the exponent is 3.
So why are exponents used in math?
Well like we talked about before, they're just convenient abbreviations for something
that would otherwise be difficult to write.
So for example if we wanted to express x being multiplied by itself 7 times, without exponents
we would have to show that multiplication as x multiplied by itself 7 times.
So one two three four five six seven times, x multiplied by itself.
But once we have exponents and we're able to use them to abbreviate this, this x multiplied
by itself 7 times just becomes x to the power of 7 or x to the 7th.
In math you always want to write things in the simplest way possible, and obviously x
to the 7 looks a lot simpler than x multiplied by itself 7 times written out like this.
Which is why we like to use exponents.
But which one of these do you think is simpler?
Is it simpler to say 2 to the 3rd or 8?
Well these are actually equal to each other because remember 2 to the 3rd is 2 times 2
times 2.
2 multiplied by itself 3 times.
Which, 2 times 2 would be 4, multiplied by this last 2 that we didn't use yet.
And then 4 times 2 is 8.
So 2 to the 3rd is the same thing as 8.
But which one would you rather see in your math problem?
Well it would definitely be the 8 because 8 is a lot simpler than 2 to the 3rd.
So exponents are really helpful when we're talking about simplifying something like x
to the 7th, but when it comes to whole numbers like this, often times it's simpler to write
out just the number rather than writing it in exponential form, because you'd much rather
have 8 than 2 to the 3rd.
So this is really the primary reason why we learn exponents in the first place and why
exponents are useful in math.
They can just make our expressions and our equations a lot simpler which will make them
easier for us to solve.
Now we're going to talk about how to simplify exponents and how to solve exponent problems.
And in order to do that, you're going to need this collection of basic exponent rules.
And there's quite a few of them, but we're going to go through each one to make sure
you understand how to use all of them.
So the first one is called the "0 rule" and this is what you use whenever the exponent
is 0.
And the rule here is that anything raised to the 0 power is equal to 1.
So in other words if I say x to the 0, because the exponent is 0 this will be equal to 1.
When I say 2 to the 0, that will be equal to 1 because the exponent is 0.
If I even have something like this, (ab) raised the 0 power, this looks more complicated,
but again because the exponent is 0 this is equal to 1.
So no matter what the base is, whether the base is x, whether it's 2, or whether it's
(ab), like this, the value is always 1 because the exponent was 0.
The only exception to this is 0^0, which honestly is a whole different discussion.
But just know that anything else raised to the 0 power is 1.
That goes for variables like x or the product of variables like (ab), or constant numbers
like 2, and even negative numbers.
For example -2 to the 0 power would also still be 1 because, regardless of the base, as long
as the exponent is 0 the value is 1.
The next rule is called the "1 rule", and just like the 0 rule, this is the rule that
you use when the exponent is 1.
So in the same way we could have x to the 1 and this is going to be equal to x, because
the 1 rule says that anything raised to the power of 1, anything where the exponent is
1, the exponent won't change the value of the base.
So the result is always just whatever you had for the base.
So x to the 1 is still just x. 2 to the 1 is still just 2, or something like this (ab)
to the 1 is still just (ab).
So notice how we always just get the base back.
And this is even true for 0 to the 1.
0 to the 1 is still 0.
This also works for negative numbers like (-2) to the 1 is still just -2.
And the reason this works is because remember the exponent tells you how many times to multiply
the base by itself.
So for example 2 to the 1st power is just telling you "Multiply 2 by itself 1 time."
And you might think that that means actually 2 times 2, right?
If I multiply 2 by itself 1 time, maybe that's 2 times 2.
But this will be multiplying 2 by itself 2 times.
This would be 2 squared.
Multiplying 2 by itself 1 time is just 2 which is why you always just get the base back when
you raise something to the power of 1.
Now the 0 rule and the 1 rule are fairly straightforward.
This next rule is where things start to get a little bit trickier.
This one is called the "power rule" and a lot of times people remember it by thinking
of "a power to a power".
And when you use this rule you'll be multiplying exponents together.
I'm going to talk a little bit later about when you add exponents together.
So with power rule, you have something like this.
For example x to the a, and then you're raising that to the power of b.
Now I know this might be confusing at first, but the next example I'm going to do is going
to make it a lot easier.
I just wanted to tell you that when you have a base x, and then an exponent a, and then
you raise that whole thing to another exponent, you multiply these powers together.
So when you multiply a times b, you're going to get ab.
So this is the same thing as x to the ab, because the exponents get multiplied together.
Here's the example that will show you why this is true.
Let's say we have 2 to the 3rd and then we're raising that to the power of 2.
If you just use the power rule for exponents to simplify this, you'll look at this and
say, "I need to multiply the exponents, so 3 times 2 is 6, which means that this is 2
to the 6th power."
But even if you didn't know power rule, here's how you would figure this out.
You would first simplify what's inside the parentheses, which goes back to your order
of operations, and I'll talk about that in a few minutes.
So you always do what's inside the parentheses first, and 2 to the 3rd power is 2 times 2
times 2, 2 multiplied by itself 3 times.
So when I do 2 times 2, right here, I get 4, and then I still have one 2 left.
So then when I say 4 times 2 I get 8.
Which means what's inside the parentheses is 8, and now I just have 8 to the 2, or 8
squared.
When I do 8 squared that's the same thing as 8 times 8, which I know is 64.
So 2 to the 3rd, squared is 64.
And what I would see is that if I do 2 to the 6th power, let's do that up here.
So 2 to the 6th power is 2 multiplied by itself 6 times.
So 2 times 2, 3, 4, 5, 6.
And here's the simple way to do this.
I can take 2 times 2 and I know this is 4.
I can take this 2 times 2 and I know that that's 4.
And I can take this 2 times 2 and I know that that's 4.
So this 2 times 2 times 2 times 2 times 2 times 2 can be simplified to 4 times 4 times
4.
Now when I do that problem, 4 times 4 is 16.
So I end up with 16 from these two, and I just have the one 4 left over.
So 16 times 4 and 16 times 4 is equal to 64.
And remember 64 was the value that we got before here, which is how we know that our
power rule worked.
Because we used power rule to simplify this to 2 to the 6th power, and when we did that
arithmetic out we got 64.
Or we just calculated it one exponent at a time without power rule and we did 2 to the
3rd to get 8, and then we took 8 squared to get 64.
And we got the same answer using both methods.
The next one is the negative exponent rule.
And this one can be a little tricky but there's actually a simple way to think about it.
So this is the rule you use when you deal with negative exponents.
For example, let's say that we have x to the -a.
So our exponent is negative because the exponent is -a.
The base is x and the exponent is -a.
The first thing you want to do whenever you have a negative exponent is immediately change
it to a positive exponent.
Because positive exponents are just easier to deal with.
So the way that you change this to a positive exponent is you think about x to the -a as
a fraction.
So before you panic here's what I mean when I say fraction.
So whenever you have something like this it's clearly not a fraction.
You can change it into a fraction just by putting this whole thing over the denominator
of 1.
Now it's a fraction but we haven't changed the value of it at all because when you make
the denominator 1 you're just dividing by 1.
But dividing by 1 doesn't change the value of anything.
For example if I take 3 divided by 1 it's still just 3.
Same thing here x to the -a all divided by 1 is still just x to the -a.
So if you don't have a fraction, you can always make it a fraction just by making the denominator
equal to 1.
Okay, so now here's what the negative exponent rule says as far as how to make this negative
exponent a positive exponent.
What you want to do is you want to take this whole thing, whatever the exponent applies
to, so you want to take the base and the exponent, and you want to move that down to the denominator.
When you do that the exponent will change from negative to a positive.
So what you get when you do that is you get the x to the a in the denominator and we were
just able to change the exponent from -a to +a because we moved it from the numerator
to the denominator, and that's how you make negative exponents positive exponents.
All you have to realize here is that we still have the 1, so we still have the 1.
And we still have a 1 in the numerator because this x to the -a is just like multiplying
1 by x to the -a.
It doesn't change that value.
So when you pull x to the -a out of the numerator and you put it into the denominator you're
not left with 0 up here, you're left with 1.
Now it's redundant to write 1x^a, because multiplying by 1 doesn't change the value.
So we really don't need this 1 down here we can just say that x to the -a is the same
thing as 1 over x^a.
So if you take another example, let's say 2 to the -3, that's always just the same thing
as 1 in the numerator and then 2 to the +3 in the denominator.
So we just move this whole thing down to the denominator in order to make the exponent
positive instead of negative.
Now here's the other interesting thing.
I can also go the opposite way, so for example if I have 1 over 2 to the -3, I have a negative
exponent in my denominator, but I can make it positive, just by moving it to the numerator.
So this is the same as 2 to the +3.
So I just moved it from the denominator up to the numerator and it made the exponent
positive.
So whenever you use the negative exponent rule just remember that if you have a negative
exponent in the numerator you can make it positive by moving it to the denominator,
and when you have a negative exponent in the denominator you can make it positive by moving
it to the numerator.
The next one is the product rule, so the product rule for exponents.
And with this one, what you're going to want to do is add your exponents.
So remember before we had the power rule, and with the power rule we multiplied our
exponents.
But with the product rule we're going to add our exponents.
So for example you use the product rule when you have something like this.
Let's say you have x squared multiplied by x cubed.
So these two things are multiplied together which we know because they're right next to
each other.
We could have a multiplication dot in there or they could be multiplied together like
this inside parentheses.
But either way we're multiplying x squared by x cubed so a problem like this one is when
you use product rule.
Notice that the bases are the same, so the base of x squared is x, and the base of x
cubed is x.
So because the bases are the same, we can multiply these together using the product
rule, and when we do that we add the exponents.
So the base stays the same but the exponents get added.
So this is raised to the exponent 2+3, which is going to be equal to x to the 5th.
If you had something like 3 squared multiplied by 3 to the 4th, you would use product rule
here too, because the bases are the same and we're multiplying these two terms together.
So because we're multiplying and because the bases are the same we can use product rule,
and the result is to keep the base the same, 3, and then add the exponents.
So 2 plus 4 and that'll give us 3 to the 6th power.
So 3 to the 6th power by the product rule is the same as 3 squared times 3 to the 4th,
because with product rule you add the exponents.
You can't use product rule if the bases are different.
So if I instead of x squared times x cubed had a squared times b cubed, I can't use the
product rule to multiply these two together, because my bases are different.
One base is a and the other base is b.
If you have different bases but the same exponent, this is a slightly different property of exponents,
but sometimes it trips people up.
If I have a squared times b squared, my bases are still different so I can't use the product
rule.
But when the exponents are the same I can rewrite this as (ab) squared.
But this is only when these terms are multiplied together.
If you have a squared plus b squared or a squared minus b squared you can't do this.
Again we're talking about the product rule where product means multiplication.
So this applies to terms that are being multiplied together.
So if we have a squared multiplied by b squared, we can put the ab together and put the exponent
outside of it.
Now this is the last one and it's similar to the product rule but this one is called
the "quotient rule".
So product rule remember said "product", which is multiplication.
Quotient rule has "quotient" and remember "quotient" means division.
Remember how with product rule we added the exponents?
Well quotient rule is where we subtract the exponents.
You use quotient rule when you have something like this.
Let's say you have x to the 4th power divided by x to the 2.
So this is a fraction with x to the 4th in the numerator and x squared in the denominator.
And in this case when you simplify this, just like product rule, because my bases are the
same, I have base x for both terms, my base will stay the same.
But the exponents simplify by taking the exponent in the numerator 4 and subtracting the exponent
in the denominator 2.
So 4 minus 2 is 2, so this is equal to x squared.
And there's another way to think about this.
Let's write out the terms that we have here.
So x to the fourth is the same as x times x times x times x and x squared is the same
as x times x.
Well remember with fractions we can always cancel terms that are common to the numerator
and denominator.
So here if we cancel out two x's from the denominator, that means we're canceling out
two x's from the numerator.
And notice then that the only thing we're left with is two factors of x in the numerator,
which we could simplify to just x squared.
So that's why the quotient rule works because you're basically just canceling common factors.
Now what about this one?
What about when you have x squared in the numerator and x to the fourth in the denominator?
Well you can follow the exact same process we did before.
Because the bases are the same, the base is still x.
Then you take the exponent in the numerator and subtract the exponent in the denominator,
so we get 2 minus 4.
Well 2 minus 4 is -2 so this is x to the -2.
This is where the negative exponent rule would come into play.
So we have x to the -2.
If we want to make that positive, we just move it to the denominator, and we say that
this is the same as 1 over x to the +2.
So 1 over x to the +2 is the result.
There are two ways to tackle this.
The first way where we take the exponent in the numerator and subtract the exponent in
the denominator, like we did in this first example.
Or we can do a little shortcut here so that we don't have to make this change at the end
in order to make the exponent positive.
What you can do instead is recognize that the exponent will be positive wherever we
have the larger exponent.
So when we have x squared over x to the fourth, 4 is larger than 2.
So what we can do then is instead of sticking with our numerator minus denominator rule,
we could put the x wherever the larger exponent exists.
The larger exponent is in the denominator so we put the exponent in the denominator,
and then for the exponent on that base we say 4 minus 2.
We always take the larger number minus the smaller number.
4 minus 2 is 2, and then we just put a 1 on the other side of the fraction.
So you can do it that way also.
Okay, so what if our bases are different?
All these examples are where the bases are the same.
If we have different bases, for example a to the 4th over b squared, we can't simplify
this because our bases are different so we're not able to apply quotient rule.
But like product rule there is that one exception where the bases can be different, but the
exponents can be the same.
So if we have a squared over b squared, like that product rule, because the 2 is applied
to the a and the b separately, we could also write this as a divided by b, and pull the
exponent outside the parentheses.
Now I know all these rules can get confusing but the one thing that I would say is that
if you're ever not sure, one of the best tricks that you can use for simplifying exponents
this way, is to write out the exponents in their long form like this.
So if you're ever not sure for example about x to the 4th over x squared, if you're supposed
to subtract the exponents or add them or multiply them, you can always write out x to the 4th
like this.
Or you can write out x squared like this.
And then you can cancel factors and reassure yourself that this is x squared.
Or for example with product rule where you have something like x squared multiplied by
x cubed, and you don't remember if you're supposed to add the exponents or multiply
them, try writing them out.
So x squared you know is x times x and you know that x to the 3rd is x times x times
x.
And then when you look at this you see that you have x multiplied by itself 5 times which
you know is x to the 5th.
So that tells you that when you have these multiplied together you're supposed to add
the exponents because 2 plus 3 is 5.
So if you're ever unsure, just try writing it out like this or like we did for this problem
and that might help you figure it out.
Now just a quick note about order of operations and PEMDAS.
So remember that order of operations tells you to start first with parentheses then exponents
and then here you have multiplication, division, addition, and subtraction.
I want to focus on the first part: parentheses then exponents.
So with exponent rules a lot of times you're going to have something inside parentheses
and then an exponent outside the parentheses.
Like in the last quotient rule example where we had (a/b) squared.
Or if you had something like 3 plus 4 cubed.
Or maybe you have something like x squared y to the 4th power.
Your order of operations tells you that you always always always want to simplify what's
inside the parentheses before you go applying the exponents.
So you always have to remember when you start an exponent problem if there's anything inside
the parentheses that you can simplify first, you definitely want to do that.
So in these two examples, this one here, and this one, there's nothing that we can simplify
inside the parentheses.
So we could then go ahead and apply the exponent and this would come out to a squared over
b squared.
This would come out to x to the 8th y to the 4th.
But this here can be simplified.
So we would definitely want to do what's inside the parentheses first.
We would say 3 plus 4 is 7 and we would get 7 and then apply the exponent of 3.
We would never want to say that this is 3 cubed plus 4 cubed or some other application
of the exponent.
We always want to simplify what's inside the parentheses first.
What about when exponents are negative?
Well we talked about the negative exponent rule before which told us that when we have
x to the -n that's the same thing as 1 over x to the +n.
We just move it to the denominator in order to make the exponent positive.
We also talked about the reverse of that, where if we have something like this, where
we have a negative exponent in the denominator, we just move it to the other side of the fraction
the numerator to make it positive.
So wherever you have a negative exponent, you can just move it to the opposite side
of the fraction and the exponent will become positive.
So that's what we do when exponents are negative, and that's the easy way to get rid of negative
exponents.
But what about when the base is negative?
This is really important.
So when I have something like negative x to the n, that means I have a negative base.
My base is -x because it's inside the parentheses.
Which means that the n applies to both the negative sign and the x.
In fact you can even think about this as -1 times x.
And when you have an exponent outside of the parentheses, the exponent is distributive,
which means it distributes to both the -1 and the x.
Or to the negative sign and the x.
So this is the same thing as -1 to the n times x to the n.
The exponent gets distributed to both the -1 and the x.
So I have a negative base where I have to distribute the exponent to that negative sign.
This however is not at all the same thing.
This is not a negative base because I don't have parentheses around my negative sign,
which means that the exponent n only applies to the x.
It's basically the same thing as saying this.
I have x to the n power, my base is a positive x, and this negative sign just gets applied
after the fact.
So let's see what that looks like.
So if for example I have (-2) squared, I do have a negative base because I have parentheses
around the -2.
So this is the same thing as saying the entire base -2 times -2, because this says multiply
-2 by itself 2 times, since the exponent is 2.
Well -2 times -2 is a positive 4 because my negative signs cancel and 2 times 2 is 4.
But if I have -2 squared, because there are no parentheses, the 2 does not get distributed
to the negative sign, which means that I basically have parentheses around the 2 squared.
I always do my operations inside the parentheses first, so 2 squared is 4 and this is equal
to -4.
And then when I take the parentheses away I get -4.
So I got a totally different answer -4 when I had no parentheses around the negative sign,
than I did +4 when I did have parentheses around the negative sign.
The last thing I want to talk about before we work an example is when exponents are fractions.
It's important to realize that fractional exponents are just a different way of writing
roots or radicals.
So for example x raised to the 1/2 power where the exponent is 1/2 is exactly the same thing
as the square root of x.
It's just a different way of writing these values but these two things are equal to one
another.
In the same way, x to the 1/3rd is equal to the 3rd root of x.
So this is the square root of x equal to x to the 1/2 power.
This is the third root of x or the cube root of x equal to x to the 1/3 power.
There's a couple really interesting things to know about fractional exponents.
First if you want to convert a fractional exponent into a root or vice-versa, you can
separate the numerator and the denominator.
So here our numerator is 1 and our denominator is 2.
The numerator always remains as an exponent on the base.
So here under our square root, we have x to the 1 because the numerator was equal to 1.
The value for the denominator becomes the root, so 2 is the special case where this
is just the square root, but here with 3 it's the cube root, which is why you see the 3
out here, because that's the denominator and this is x to the 1st power inside the root.
So therefore x to the 1/4 is the same thing as the 4th root of x to the 1st.
Which means that if we have x to the 2/3, and we want to convert that to a root, we
get our square root here, our x goes underneath it, the numerator is a 2, so that stays on
the base, this is x squared, and the denominator is a 3, so this is the third root.
So that's how we convert between roots and fractional exponents but there's one other
important thing to say about exponents which are fractions.
And that is that we have to remember the power rule from earlier.
Remember the power rule said that if we had x to the a and then raised to the b, that
we multiply these exponents by each other and this is x to the a times b.
And these are the same thing.
With fractional exponents we have to remember that the fraction 2/3 is the same thing as
2 times 1/3.
So we could rewrite x to the 2/3 as x squared and then raise that whole thing to the 1/3.
So these two values are exactly the same because by the power rule we multiply these two exponents
together to simplify this, and we get x to the 2 times 1/3.
Well 2 times 1/3 is 2/3.
So we get x to the 2/3.
So if it helps you to simplify your problem, which it often does, you can always separate
exponents like this by putting the numerator here on the inside and just keeping the denominator
here on the outside.
So let's put everything we've learned into practice on one really tough example.
Let's say that we have the fraction -4 x cubed y to the -1 and then we square that whole
thing, and that's multiplied by 5 x cubed, y to the -2, and that whole thing is raised
to the 0 power.
And then we want to divide that by 2 x to the 4th y, all raised to the 3rd power.
Remember that we have these rules and you always want to apply the rules in this order.
So first the 0 rule, then the 1 rule, then power rule, then negative exponent rule, product
rule, and quotient rule.
Think about it as an order of operations specifically for exponents.
Now speaking of order of operations, remember we always want to try to simplify inside of
our parentheses before we go into applying our exponent rules.
Well for us there's nothing to be simplified inside the parentheses.
We can't simplify -4x^3y^(-1), we can't simplify 5x^3y^(-2), and we can't simplify 2x^4y.
So we can't do anything inside the parentheses which means that we can now move on to applying
our exponent rules.
So the first thing we want to do is the 0 rule and we do that first because that can
knock out something really quickly.
Remember that when we raise something to the 0 power that it's equal to 1.
So what we see here is that we have this whole value in parentheses (5x^3y^(-2)), and the
whole thing is raised to the 0 power.
Remember from before that it wouldn't matter what was inside these parentheses, we are
raising the entire thing to the 0, which means that that whole term is going to be equal
to 1.
So what we can do is we can say that this whole thing is equal to 1.
But remember that multiplying something by 1 doesn't change its value at all.
So multiplying by 1 here isn't going to have any effect on the rest of the numerator, so
we can get rid of that completely.
Now there's nothing else that's raised to the 0 power so we want to go on to the second
rule, the 1 rule and see if anything is raised to the power of 1.
We've got this y^(-1) term but we're only looking for things that are raised to the
power of +1, that's the 1 rule.
So we can't apply the 1 rule to y^(-1).
We could only apply it if it were y^(+1).
So there's nothing to simplify there, and we can say that we've done this rule, the
0 rule, and the 1 rule.
Now what about the power rule?
Remember that the power rule is "a power to a power", or raising one exponent to another
exponent.
Well in our case that means distributing these exponents on the outside across everything
inside the parentheses.
Remember that exponents are distributive when we have all multiplication inside our parentheses,
and here we do.
We have -4 multiplied by x^3 multiplied by y^(-1), so it's all multiplication and we
can distribute the 2 across each term.
Same thing here we have 2 multiplied by x^4 multiplied by y.
It's all multiplication so we can distribute the 3.
In order to do that we want to remember that anything that doesn't have an exponent, like
this -4 and the 2, the implied exponent is a 1.
That kind of comes from the 1 rule.
So now when we apply the power rule, remember that the power rule tells us to multiply exponents.
Which means we want to multiply this outside exponent across all the exponents inside the
parentheses.
So here our negative sign is inside the parentheses, which means the -4 is a negative base and
that negative sign gets included.
So we want to say -4 to the 1 times 2, multiplying those exponents, multiplied by x to the 3
times 2 multiplied by y to the negative 1 times 2.
And then in the denominator we want to say 2 to the 1 times 3, times x to the 4 times
3 times y to the 1, because again we have an implied 1 there, times 3.
So when we simplify, we get (-4)^2, we get x^6 because 3 times 2 is 6, -1 times 2 is
-2 so we get y^(-2), and then in the denominator we get 2 to the 1 times 3, 1 times 3 is 3,
so 2^3, 4 times 3 is 12 so we get x^12 and 1 times 3 is 3, so we get y^3.
So that's power rule applied.
Now we want to apply the negative exponent rule which remember we do for any of the negative
exponents that we have by moving that term from the numerator to the denominator or vice
versa.
So in this case our only negative exponent is here with y^(-2).
So we just want to move that to the denominator in order to make it positive.
So we'll have (-4)^2 x^6, and then we're going to move that y^(-2) to the denominator.
So we have 2^3 x^12, y^3, and then here our y now to the +2, because it's in the denominator.
And that's the negative exponent rule because now all of our exponents are positive.
Then we move on to the product rule, which tells us that when we have like bases, we
add the exponents.
So here we have y^3 multiplied by y^2.
We know that product rule allows us to add those exponents together, so this becomes
y to the 3+2 or y^5.
So we can go ahead and write this as y^5.
Now there's no other values in the numerator or the denominator where we can use product
rule, so we finished that one, but now we have quotient rule.
And with quotient rule, remember that when we have like bases across the numerator and
denominator, we subtract the exponents.
So here we have x^6 in the numerator and x^12 in the denominator.
Remember that the term is going to stay with the largest exponent.
So because 12 is greater than 6, that means that we're going to end up with a positive
exponent in the denominator.
So what we want to do is subtract the value in the numerator from the value in the denominator.
So let's move over our y^5 a little bit, and now we're going to want to say 12 minus 6
here in the denominator because the exponent in the denominator is larger.
And that gets rid of x^6 in the numerator.
So 12 minus 6 then is 6, so we're left with x^6 in the denominator.
Now that's quotient rule done.
We've done all of our exponent rules.
We just need to simplify our constants.
So here we have -4 squared.
The -4 is inside the parentheses, so this is a negative base and we do have to apply
the exponent to the negative sign.
So this is telling us to multiply -4 by itself 2 times.
So this is the same thing as -4 times -4 which is positive 16 because the negative signs
cancel and 4 times 4 is 16.
And then in our denominator we have 2 to the 3rd, which remember is 2 times 2 times 2,
and if we have 2 times 2 times 2, we know that 2 times 2 is 4, 4 times 2 is 8.
So we get 8.
And then we have x^6 y^5.
And remember whenever you get to the end, you always want to make sure that there's
nothing else you can reduce.
In this case we see that we have 16 in the numerator and 8 in the denominator, which
means we can factor out an 8.
So we know that 16 is the same thing as 8 times 2 and because we have an 8 in the numerator
and the denominator, and because everything in the numerator is multiplied together and
everything in the denominator is multiplied together, we can cancel those 8s, and we're
just left with the 2 in the numerator.
So then our final answer is 2 divided by x^6 y^5.
I hope that video helped you, and if it did, hit that like button, make sure to subscribe,
and I'll see you in the next video.
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