WEBVTT mathematics/pre-calculus/selhorst-jones
00:00:00.000 --> 00:00:02.300
Hi--welcome back to Educator.com.
00:00:02.300 --> 00:00:05.500
Today, we are going to talk about understanding exponents.
00:00:05.500 --> 00:00:12.300
At this point, we are quite used to using exponents; we have seen them a bunch, and we just did them a whole lot when we were working with polynomials.
00:00:12.300 --> 00:00:16.600
We know that an exponent is just a shorthand way to express repeated multiplication.
00:00:16.600 --> 00:00:26.000
For example, if we had 3⁷, that would be a way of just saying 3 times 3 times 3 times 3 times 3 times 3 times 3.
00:00:26.000 --> 00:00:35.600
That is 3, multiplied by itself 7 times; so the number of times it multiplies is what the exponent is; that is what it represents.
00:00:35.600 --> 00:00:44.400
x² is x times x; x³ is x times x times x (three x's); x⁴ would be four x's multiplied together; etc.
00:00:44.400 --> 00:00:46.800
So, that is the idea of exponentiation.
00:00:46.800 --> 00:00:52.400
It is clear how this process works when the exponents are positive integers; it is just multiplied by itself that many times.
00:00:52.400 --> 00:00:58.600
But if we wanted to expand to any real number--what if we wanted to be able to exponentiate to any real number at all?
00:00:58.600 --> 00:01:06.400
If we wanted to know things like 3⁰, 3^7/8, 3^-5, or 3^√2, how can we deal with that?
00:01:06.400 --> 00:01:10.400
This lesson will show us how to work with any kind of exponent--any real number.
00:01:10.400 --> 00:01:17.000
You may have seen rules for this stuff in a previous math class; but we are also going to work through an understanding of why these rules are true.
00:01:17.000 --> 00:01:20.800
Even if you remember what the rules are for what 3⁰ is, for instance,
00:01:20.800 --> 00:01:25.100
you might not have a good grasp of why 3⁰ is what it is.
00:01:25.100 --> 00:01:29.400
And so, this is the lesson that we are actually going to see how we build these rules, where they come from,
00:01:29.400 --> 00:01:34.500
why we can trust in them, and also how we can make them ourselves, if we ever forget the rules.
00:01:34.500 --> 00:01:39.900
If we forget the long summary list of all the rules for exponents, we will be able to just go back to our workshop
00:01:39.900 --> 00:01:47.100
and work it out, even in the middle of an exam; it won't take that long for us to figure out, if we have a sense of how we get these things.
00:01:47.100 --> 00:01:54.600
All right, at heart, exponentiation is this idea of repeated multiplication; that is the basic, fundamental idea.
00:01:54.600 --> 00:02:02.900
By definition, for any number x and any positive integer a, x^a is equal to x, multiplied by itself a times.
00:02:02.900 --> 00:02:06.600
This is the key idea behind all of our coming rules for exponents.
00:02:06.600 --> 00:02:12.100
If we start with this idea, and we just hold it as a fundamental truth and see all the places that it can take us,
00:02:12.100 --> 00:02:16.900
we will be able to get all of these other cool rules that will explain how it will work for any real number.
00:02:16.900 --> 00:02:21.600
We take this idea; we say that we believe in this; and then we move forward and try to figure out
00:02:21.600 --> 00:02:27.400
what else has to be true if this idea right here is always going to work out.
00:02:27.400 --> 00:02:33.300
Let's see: the first thing we can figure out: with this idea in mind, we can consider what happens when we multiply
00:02:33.300 --> 00:02:39.900
(that should be "multiply," not "multiple") some x^a and x^b.
00:02:39.900 --> 00:02:45.500
By definition, if we have x^a times x^b, then that means we have x^a,
00:02:45.500 --> 00:02:52.300
so we have x multiplied by itself a times; and x^b is x multiplied by itself b times.
00:02:52.300 --> 00:02:59.500
Now, that means we have some number of x's there and some number of x's there; we have a x's on one side and b x's on the other side.
00:02:59.500 --> 00:03:10.100
So, how many is it together? Well, let's say, if I had a pile of 5 rocks over here, and I had a pile of 12 rocks over here, then in total, I would have 17 rocks.
00:03:10.100 --> 00:03:15.900
If I had 3 rocks here and 4 rocks here, then I would have 7 rocks total.
00:03:15.900 --> 00:03:26.300
So, if we have a rocks and b rocks here, then together that is going to be a + b rocks (or in this case, x's).
00:03:26.300 --> 00:03:31.500
So, we put them together, and we see that we have a + b many x's showing up.
00:03:31.500 --> 00:03:37.200
Thus, x^a times x^b equals x^a + b.
00:03:37.200 --> 00:03:46.600
This is another really fundamental property, and this is what is going to allow us to explore a lot of things in exponentiating with real numbers.
00:03:46.600 --> 00:03:49.900
We can look at exponentiation acting on top of exponentiation.
00:03:49.900 --> 00:03:57.200
If we have x^a, and then raise that to the b, then that means we have x^a multiplied by itself b times.
00:03:57.200 --> 00:04:01.000
So, this will give us a total of a times b x's; why is that the case?
00:04:01.000 --> 00:04:11.400
Well, imagine if, in each box, there are 5 rocks; well, if we have 3 of these boxes, then it is 3 times 5.
00:04:11.400 --> 00:04:18.100
So, if we have a rocks in each box, a x's in each of these boxes (each of these x^a's),
00:04:18.100 --> 00:04:23.000
and we have b many of them, then it must be a times b of them in total.
00:04:23.000 --> 00:04:28.400
Thus, (x^a)^b is equal to x^ab.
00:04:28.400 --> 00:04:31.700
We just multiply the two exponents in that case.
00:04:31.700 --> 00:04:36.600
We can also consider what happens if we have two different numbers, each raised to the same exponent.
00:04:36.600 --> 00:04:43.000
If we have x^a times y^a, then we have a many x's and a many y's.
00:04:43.000 --> 00:04:49.800
But we can also shuffle things around: xy is the same thing as yx; we are pretty confident in this.
00:04:49.800 --> 00:04:54.100
That is one of the cool things about working with the real numbers--that they are commutative.
00:04:54.100 --> 00:04:59.500
They are allowed to swap their spaces; 2 times 3 is the exact same thing as 3 times 2.
00:04:59.500 --> 00:05:05.700
So, x times y is the same thing as y times x, which also means that, if we have xx times yy,
00:05:05.700 --> 00:05:10.600
if we feel like it (because we are allowed to commute--we are allowed to swap the locations in multiplication),
00:05:10.600 --> 00:05:15.000
then we could switch this to xyxy, which is exactly what we do here.
00:05:15.000 --> 00:05:23.600
We take these x's and these y's, and we file them together: one x here, one y here, the next x here, and the next y here.
00:05:23.600 --> 00:05:29.200
So, we will have xyxy; that will show up a total of a times, as well.
00:05:29.200 --> 00:05:37.500
x^a times y^a...if we want, we can write it as (xy), that quantity, all raised to the a.
00:05:37.500 --> 00:05:42.400
Now, let's try to figure out what happens when we raise a number to the 0--our first sort of difficult question.
00:05:42.400 --> 00:05:53.800
x⁰ equals what? First, we can write x = x any time we want; just definitionally, that is the idea of equality; you are equal to yourself.
00:05:53.800 --> 00:05:58.200
So, we can write x = x as x¹ = x¹.
00:05:58.200 --> 00:06:02.800
We can put it to an exponent of 1, because that just means that it is itself, just multiplied once--
00:06:02.800 --> 00:06:05.400
it is just there by itself, because there is only one of them.
00:06:05.400 --> 00:06:09.500
So, there is nothing wrong with writing x = x as x¹ = x¹.
00:06:09.500 --> 00:06:16.000
But not only that--we know that 1 is equal to 1 + 0; 1 + 0 is just 1.
00:06:16.000 --> 00:06:23.200
So, if we want, we can take this 1 right here, and we will substitute it for 1 + 0; and we have x^1 + 0.
00:06:23.200 --> 00:06:28.100
And then, we will just sort of knock out this one, and leave it as x, just so it is easier to read what is going on.
00:06:28.100 --> 00:06:31.800
So, we have x^1 + 0 = x.
00:06:31.800 --> 00:06:36.500
This means we have a 0 on the field that we can play with.
00:06:36.500 --> 00:06:42.500
By our new property, x^a + b equals x^a times x^b; it is our most basic property.
00:06:42.500 --> 00:06:47.200
Then, we can break this up, and we can take the 1 and separate it from the 0.
00:06:47.200 --> 00:06:55.700
So, we will have x¹ and x⁰; and we will just write x¹ as just x by itself, since that is what it is.
00:06:55.700 --> 00:07:08.500
We have x(x⁰) = x; now, as long as x is not equal to 0 (if x is equal to 0, we can't really divide by it easily),
00:07:08.500 --> 00:07:13.100
we divide by x; we cancel out the x's on both sides, since they show up on both sides.
00:07:13.100 --> 00:07:17.600
They disappear, and we are left with x⁰ = 1.
00:07:17.600 --> 00:07:25.400
There we go: thus, any number, as long as it isn't 0, raised to the 0, becomes 1.
00:07:25.400 --> 00:07:29.400
So, we take any number at all; we raise it to the 0; it is going to become 1.
00:07:29.400 --> 00:07:41.500
5⁰ = 1; (-52)⁰ = 1; 47 million to the 0--you guessed it--equals 1.
00:07:41.500 --> 00:07:46.300
So, whatever we take, if we raise it to the 0, it becomes 1.
00:07:46.300 --> 00:07:49.000
Next, what happens when we raise something to a negative number?
00:07:49.000 --> 00:07:52.000
For ease, let's just figure out what happens with x^-1.
00:07:52.000 --> 00:07:59.100
We begin similarly: x¹ = x¹; you can't stop me from saying that, just because it is the same thing on both sides.
00:07:59.100 --> 00:08:10.400
And then, we can also say that 1 = 2 - 1; 2 minus 1 equals 1, so if we want, we can substitute in on this side here; and we have x^2 - 1 = x.
00:08:10.400 --> 00:08:15.900
Now, notice: we have a -1 on the field; so we use that property; we break it apart.
00:08:15.900 --> 00:08:27.600
And so, by our property, x^a + b becomes x^a times x^b, we have x² times x^-1.
00:08:27.600 --> 00:08:35.000
So, x² times x^-1 = x; now, we can divide by x², once again, as long as x is not equal to 0.
00:08:35.000 --> 00:08:37.500
If we have that, then things get troublesome.
00:08:37.500 --> 00:08:44.900
But if we divide by x², the x² goes from here over to here.
00:08:44.900 --> 00:08:54.500
And so, we get x^-1 = x/x²; and so, x/x² cancels the x on top
00:08:54.500 --> 00:08:58.200
and turns this to a 1, and we are left with = 1/x.
00:08:58.200 --> 00:09:06.800
So, x^-1 = 1/x; thus, any number, as long as it isn't 0, raised to a negative, flips to its reciprocal.
00:09:06.800 --> 00:09:15.500
If we have some number, and we raise it to a negative, we will get that number, flipped into the reciprocal format.
00:09:15.500 --> 00:09:22.900
5^-1 = 1/5...so, what if we want to know what x^-a is for any a at all?
00:09:22.900 --> 00:09:33.700
Well, from the other work that we just did, we know that x^-1 to the a is x to the -1 times a, so we can write it in that way, as well.
00:09:33.700 --> 00:09:40.800
And so, we have 1/x^a = 1/x^a.
00:09:40.800 --> 00:09:45.500
A negative in our exponent causes it to flip to its reciprocal format.
00:09:45.500 --> 00:09:50.300
But it will still keep whatever that original exponent number was, as well.
00:09:50.300 --> 00:09:59.400
The number will go with it, but the negative is a flip; so negatives flip, but this number that we are exponentiating to will still stay with it.
00:09:59.400 --> 00:10:04.500
With this idea, we can consider if we had a fraction raised to the -1--not just a number, but a whole fraction.
00:10:04.500 --> 00:10:09.900
(x/y)^-1...well, you can't stop me from separating that into x times 1 over y.
00:10:09.900 --> 00:10:15.400
And then, we can distribute that -1; remember, xy to the a is equal to x^a times y^a.
00:10:15.400 --> 00:10:23.000
So, that -1 will go onto both the x and the 1/y; so we have x^-1 times (1/y)^-1.
00:10:23.000 --> 00:10:31.600
x^-1 flips to 1/x; (1/y) will flip to y/1, or just y; and so we have y/x.
00:10:31.600 --> 00:10:37.700
So, negative exponents flip fractions; if we have a negative exponent, we flip whatever it is,
00:10:37.700 --> 00:10:42.600
whether it is a fraction or a number--we flip to the reciprocal; great.
00:10:42.600 --> 00:10:47.400
We can also look at if we have powers in the numerator and the denominator with the same base.
00:10:47.400 --> 00:10:51.800
The base is just the thing that is having that exponentiation happening to it.
00:10:51.800 --> 00:10:55.700
So, x is our base in almost all of these examples.
00:10:55.700 --> 00:11:02.400
So, x^a over x^b...well, we can separate that into x^a times (1/x)^b.
00:11:02.400 --> 00:11:08.500
We have x^a on top, so we separate that: x^a times 1/x, and that whole thing to the b.
00:11:08.500 --> 00:11:14.700
That is equal to x^a times x^-b, because 1/x is equal to x^-1.
00:11:14.700 --> 00:11:21.300
So, 1/x to the b is equal to x^-b; and now, x^a times x^-b
00:11:21.300 --> 00:11:24.400
combines through addition, which, in this case, will become subtraction.
00:11:24.400 --> 00:11:30.900
So, we have x^a - b; thus, the denominator's power subtracts from the numerator's power.
00:11:30.900 --> 00:11:33.800
That is another thing that we have gotten out of this.
00:11:33.800 --> 00:11:36.000
Finally, what happens when we raise a number to a fraction?
00:11:36.000 --> 00:11:41.000
For ease, let's look at just x^1/2 first; that will make it easier to understand.
00:11:41.000 --> 00:11:43.900
Once again, we start from the same place: x¹ = x¹.
00:11:43.900 --> 00:11:50.200
And like usual, we want to bring 1/2 to bear; so we notice that 1 = 1/2 + 1/2--it is as simple as that.
00:11:50.200 --> 00:11:59.400
So now, we can substitute it; we swap this in here, and we have x^1/2 + 1/2 = x.
00:11:59.400 --> 00:12:05.300
Now, notice: we can use our property x^a + b = x^a times x^b,
00:12:05.300 --> 00:12:11.600
the usual property we have been using, to separate this into x^1/2 times x^1/2 = x.
00:12:11.600 --> 00:12:15.700
We already have a name for that--square root.
00:12:15.700 --> 00:12:22.200
√x times √x = x; that is the idea behind square root.
00:12:22.200 --> 00:12:29.100
The definition of square root is some number that, when you multiply it by itself, becomes the number you took the square root of.
00:12:29.100 --> 00:12:34.800
So, √x times √x...√x is just some number that, when you multiply it by itself, becomes x.
00:12:34.800 --> 00:12:43.300
So, if x^1/2 times x^1/2 is equal to x, then that must be the same thing as √x,
00:12:43.300 --> 00:12:48.500
because it does the exact same property--itself times itself becomes x.
00:12:48.500 --> 00:12:53.600
We already have a name for that: we call that square root.
00:12:53.600 --> 00:13:00.100
With this property, we see that x^1/2 = √x; they are equivalent.
00:13:00.100 --> 00:13:04.000
We can expand this, by similar logic, to x^1/n.
00:13:04.000 --> 00:13:11.200
x^1/n is the same thing as saying that x^1/n times itself n times is equal to x,
00:13:11.200 --> 00:13:16.700
because we have that n many 1/n's is equal to 1.
00:13:16.700 --> 00:13:23.500
So, if we combine x^1/n times x^1/n, we will be adding 1/n to itself n times, which is equal to 1.
00:13:23.500 --> 00:13:29.900
So, x^1/n times itself n times is equal to our x, by the same logic that we split up x^1/2.
00:13:29.900 --> 00:13:38.700
By definition, the nth root of a number is something that, when it multiplies by itself n times...we get the original number.
00:13:38.700 --> 00:13:45.300
The cube root of something, the third root of something, is a number such that, when it multiplies by itself 3 times, we get our original number.
00:13:45.300 --> 00:13:50.700
The nth root of something is a number such that, when it multiplies by itself n times, we get the original number.
00:13:50.700 --> 00:13:54.900
Well, look: we are multiplying it by itself n times; we are getting that original number out of it;
00:13:54.900 --> 00:14:00.500
so it must be that x^1/n is equal to the nth root of x.
00:14:00.500 --> 00:14:03.300
With this idea in mind, we can use any rational number that we want at all.
00:14:03.300 --> 00:14:13.200
We have x^a/b; we can separate that into (x^a)^1/b.
00:14:13.200 --> 00:14:19.700
And since we had just had this thing here, 1/n is the same thing as nth root, so we have 1/b becoming ^b√.
00:14:19.700 --> 00:14:25.200
We have the ^b√x^a, the bth root of x^a.
00:14:25.200 --> 00:14:26.900
We are just mixing the two properties.
00:14:26.900 --> 00:14:31.900
The numerator is normal exponentiation, just multiplying by itself, like we normally would expect.
00:14:31.900 --> 00:14:38.800
And then, the denominator takes a root; it takes that bth root, because it is 1/b.
00:14:38.800 --> 00:14:42.100
At this point, we have a lot of different rules, and we can see a summary.
00:14:42.100 --> 00:14:51.100
Our first, foremost, most fundamental rule of all is this idea right here, x^a times x^b = x^a + b.
00:14:51.100 --> 00:14:57.800
From there, we are able to figure out all of these other rules: (x^a)^b is equal to multiplying the two together.
00:14:57.800 --> 00:15:03.700
So, if we have two different exponents raised on one thing, it multiplies them together.
00:15:03.700 --> 00:15:10.100
If we have x^a times y^a, we can combine that to just having xy to one a.
00:15:10.100 --> 00:15:17.000
x⁰ is always equal to 1, so if you are raising to the 0, you always come out to being 1,
00:15:17.000 --> 00:15:20.100
as long as we are not dealing with x equal to 0, which we won't address.
00:15:20.100 --> 00:15:25.000
x^-a = 1/x^a; we flip with negative exponents.
00:15:25.000 --> 00:15:28.700
If you have a negative here, then you flip down to the bottom.
00:15:28.700 --> 00:15:37.200
If we have a fraction with a negative, then the whole fraction flips to its reciprocal; we have y over x to the a.
00:15:37.200 --> 00:15:43.100
And we also see that x^a divided by x^b becomes x^a - b.
00:15:43.100 --> 00:15:49.500
Finally, our nth root stuff: x^1/n is equal to the nth root of x.
00:15:49.500 --> 00:15:53.400
x^a/b is equal to the bth root of x^a,
00:15:53.400 --> 00:15:58.400
which is the same thing as the bth root of x, to the a, because as opposed to splitting it...
00:15:58.400 --> 00:16:08.000
we can split this as 1/b to the a, which we would see as ^b√x, all raised to the a.
00:16:08.000 --> 00:16:09.600
And that is where we are getting that.
00:16:09.600 --> 00:16:11.200
So, there are two different ways of looking at it.
00:16:11.200 --> 00:16:14.700
Sometimes it will be more convenient to have the bth root of x^a.
00:16:14.700 --> 00:16:18.900
Other times, it will be more convenient to have the bth root of x, all raised to the a.
00:16:18.900 --> 00:16:20.500
It will depend on the specific problem.
00:16:20.500 --> 00:16:24.500
Remember: I really want you to take away this idea here.
00:16:24.500 --> 00:16:36.800
If you forget any of these rules, you can figure them out from this fundamental, basic idea: x^a times x^b equals x^a + b.
00:16:36.800 --> 00:16:40.700
You just have to come up with some creative way to get the thing that you are trying to figure out,
00:16:40.700 --> 00:16:46.200
whether it is fractions, whether it is 0, whether it is negative numbers--
00:16:46.200 --> 00:16:51.400
you figure out some creative way to get 0, -1, 1/2, 1/n, something like that, to show up.
00:16:51.400 --> 00:16:54.400
And then, you look at it, and you say, "Oh, I see--that is what it is!"
00:16:54.400 --> 00:16:59.300
And so, even if you forget this in the middle of an exam--some place where you can't go and look it up in a book--
00:16:59.300 --> 00:17:02.000
you can figure this out on your own; it is not that hard.
00:17:02.000 --> 00:17:07.700
And having worked through it, and understanding how we are getting this, it is that much more likely to stick in your brain.
00:17:07.700 --> 00:17:14.400
I know it seems like a lot of rules; but once you start using them, and you get used to using them, they will stick in your head.
00:17:14.400 --> 00:17:19.600
And as long as you remember this one, you can ultimately get back anything that you have forgotten by accident.
00:17:19.600 --> 00:17:23.300
All right, the final idea: what if we want to raise to an irrational?
00:17:23.300 --> 00:17:28.200
So far, we have actually only discussed exponentiation using rational numbers.
00:17:28.200 --> 00:17:30.100
That is the only thing that we have technically dealt with.
00:17:30.100 --> 00:17:37.300
We have a/b for any a and b, but we haven't dealt with if it can't be expressed as a/b, like √2.
00:17:37.300 --> 00:17:39.500
So, what if we want to raise something to an irrational number?
00:17:39.500 --> 00:17:47.200
Let's say we want to look at 3^√2; notice: if we want to, we can look at as many places of √2 as we want.
00:17:47.200 --> 00:17:51.900
We can figure out that √2 is equal to 1.41421356...
00:17:51.900 --> 00:17:55.100
and it will just keep marching on forever, because it is an irrational number,
00:17:55.100 --> 00:18:02.100
so its decimal expansion goes on forever, never repeating, always changing, constantly going on forever.
00:18:02.100 --> 00:18:05.300
But we can figure out what that is.
00:18:05.300 --> 00:18:08.800
Furthermore, we know how to exponentiate to any rational number.
00:18:08.800 --> 00:18:13.700
So, we can raise to any decimal, because any decimal is actually something that we can express as a rational.
00:18:13.700 --> 00:18:20.200
For example, 1.4: if we want to, we can express that as 14 divided by 10.
00:18:20.200 --> 00:18:30.200
1.414: if we wanted to, we could express that as 1414 divided by 1000.
00:18:30.200 --> 00:18:34.400
All right, so we can do any of these based on all of the work that we just had.
00:18:34.400 --> 00:18:42.200
We could do 3^1.4 as 14/10, 3^1.414 as 1414/1000...
00:18:42.200 --> 00:18:46.100
we see that the work we have just done gives us a way to figure these things out.
00:18:46.100 --> 00:18:50.000
Of course, it would be very difficult for us to do these by hand, but there are methods to do these things.
00:18:50.000 --> 00:18:54.400
We could do it by hand, but we will leave it to the calculators, since they can do it so much faster.
00:18:54.400 --> 00:19:00.100
We can use a calculator and get this done so much faster, because they have already been programmed with how these methods work.
00:19:00.100 --> 00:19:06.500
So, we can take these various things and see: 3^1.4 becomes 4.6555.
00:19:06.500 --> 00:19:20.200
3^1.414 becomes 4.7276; 3^1.41421 becomes 4.7287; 3^1.4142135 becomes 4.7288.
00:19:20.200 --> 00:19:29.300
So, notice: as we use more and more of these decimals, we see the exponentiation, this 3^√2, sort of stabilize to a single thing.
00:19:29.300 --> 00:19:35.300
The 4 always gets used; the 7 always gets used; the 2 always gets used; the 8 always gets used.
00:19:35.300 --> 00:19:40.700
We see that it is becoming more and more and more stable--that we are seeing more and more of these decimal places show up,
00:19:40.700 --> 00:19:44.100
and they are not going to change--they are going to stay there forever.
00:19:44.100 --> 00:19:49.200
So, while we can't get the whole number all at once (it is going to end up being irrational,
00:19:49.200 --> 00:19:57.000
so it is going to also have a decimal expansion that continues forever, constantly changing), it is stabilizing to something.
00:19:57.000 --> 00:20:04.900
So, we can get this idea that, while we can't write it down on paper (because it would require an infinite amount of paper), the number does exist.
00:20:04.900 --> 00:20:09.100
And so, we can get as many decimals as we need for whatever our use is.
00:20:09.100 --> 00:20:15.500
So, we won't formally define this: but we see that irrational exponents make sense, because we are stabilizing to some number.
00:20:15.500 --> 00:20:21.500
As long as we use lots and lots of decimals when we calculate it out, 3 to the lots and lots of decimals,
00:20:21.500 --> 00:20:24.400
from what we were originally trying to use as our irrational number,
00:20:24.400 --> 00:20:27.800
we will be able to get something that is a very, very close approximation
00:20:27.800 --> 00:20:33.100
to the exact number that we are trying to strive towards, but won't ever be able to perfectly reach.
00:20:33.100 --> 00:20:36.000
All right, we are ready for some examples.
00:20:36.000 --> 00:20:40.300
Evaluate 8/27, all raised to the -2/3.
00:20:40.300 --> 00:20:43.700
With many of these examples, there are actually going to be multiple ways that we could approach it.
00:20:43.700 --> 00:20:47.300
So, I will try to show you the various ways that you could go about it.
00:20:47.300 --> 00:20:52.800
(8/27)^-2/3: the first thing I would do is see that we have this negative sign.
00:20:52.800 --> 00:21:00.000
So, I am going to flip and get rid of that negative: we have (27/8)^2/3.
00:21:00.000 --> 00:21:07.700
That equals...now we have 2 and thirds, so I would put the third into nth roots on both of them.
00:21:07.700 --> 00:21:12.500
We have the third root, the cube root, of 27, and the cube root of 8.
00:21:12.500 --> 00:21:17.200
And so, we now no longer have that dividing by 3 to worry about.
00:21:17.200 --> 00:21:21.600
But we still have the squared, because we didn't get rid of it by putting anything out there.
00:21:21.600 --> 00:21:32.200
So, the cube root of 27: 3 times 3 times 3 is 27, so we have 3; 2 times 2 times 2 is 8, so we have 2.
00:21:32.200 --> 00:21:39.000
That is all raised to the 2, once again; so that is 3²/2².
00:21:39.000 --> 00:21:44.400
It gets distributed: 9/4, and there is our answer.
00:21:44.400 --> 00:21:46.200
But there are also other ways that we could have done this.
00:21:46.200 --> 00:21:58.000
We could have seen this as (8/27)^-2/3, and we could have gone about this as flipping to (27/8)^2/3.
00:21:58.000 --> 00:22:07.300
And we could then put this as [(27²)/(8²)]^1/3; and that is going to be kind of difficult for us to do.
00:22:07.300 --> 00:22:14.700
So, we could do this with a calculator; and then we could take the cube root of 27² over 8².
00:22:14.700 --> 00:22:18.100
And that would eventually simplify out to 9/4.
00:22:18.100 --> 00:22:20.000
But that would be very difficult to do by hand.
00:22:20.000 --> 00:22:23.600
But notice that we can do the cube root of 27, and we can do the cube root of 8.
00:22:23.600 --> 00:22:26.400
So, this is probably the much easier way to do this.
00:22:26.400 --> 00:22:36.200
Furthermore, we could even go about this by taking this as 8/27; we could put this as -2/3 on 8, and then 27^-2/3.
00:22:36.200 --> 00:22:42.000
And then, since they are both negative, they would flip into 27^2/3 over 8^2/3.
00:22:42.000 --> 00:22:48.000
So, we would have 27^2/3 and 8^2/3; and then, we could, once again,
00:22:48.000 --> 00:22:52.800
do either this method here or this method here.
00:22:52.800 --> 00:23:00.600
At this point, I think pretty clearly that this here is our best bet--the easiest way to do it--
00:23:00.600 --> 00:23:08.700
where we go through this method, because we see, "8...27...I am going to have to deal with cube roots."
00:23:08.700 --> 00:23:15.500
8 and 27 are things I can easily take a cube root of, so I am going to do cube root first, then square.
00:23:15.500 --> 00:23:18.200
And I will also get rid of that negative, as just a first step.
00:23:18.200 --> 00:23:22.100
You can do these things in many different orders, because the rules all work together.
00:23:22.100 --> 00:23:26.800
But you will want to get a sense, and as you work on more examples, you will get a sense,
00:23:26.800 --> 00:23:31.600
of "Oh, the way that will make this problem easiest is for me to go through like this."
00:23:31.600 --> 00:23:34.200
And so, you will develop an intuition about it.
00:23:34.200 --> 00:23:38.100
And even if you end up going in a way that is not the easiest, it will still work out.
00:23:38.100 --> 00:23:41.300
It just might require using a calculator or require a little extra effort.
00:23:41.300 --> 00:23:51.400
All right, the next one: Simplify (x²/z)^-2 times (x²y³z^-1)³/y⁸.
00:23:51.400 --> 00:23:54.600
All right, the first thing I would do is deal with the negatives, once again.
00:23:54.600 --> 00:24:01.300
Usually that is easiest; so this will become z/x², all raised to the now positive 2,
00:24:01.300 --> 00:24:08.800
times...and let's distribute this 3; the 3 will go onto the x², the y³, and the z^-1.
00:24:08.800 --> 00:24:22.900
So, it is x to the 2 times 3, because we have exponentiation on exponentiation; y to the 3 times 3, z to the -1 times 3, all divided by y⁸.
00:24:22.900 --> 00:24:31.900
We can deal with this squared, and we get z² over...that 2 also distributes onto the top and to the bottom,
00:24:31.900 --> 00:24:37.300
so x² squared is x² times x², or x⁴.
00:24:37.300 --> 00:24:40.500
Also, it is x to the 2 times 2--another way of looking at it.
00:24:40.500 --> 00:24:51.400
Times x⁶y⁹z^-3, all over y⁸...
00:24:51.400 --> 00:24:56.200
At this point, we see that we can cancel out the y⁸, and this becomes y¹,
00:24:56.200 --> 00:25:02.500
which we could also look at as y^9 - 8, because we have one on top and one on bottom,
00:25:02.500 --> 00:25:08.000
which would also become y¹; so there are various ways to do this.
00:25:08.000 --> 00:25:20.100
z² times x⁶ times...I will move that over...let's put all of our variables so that they are near their similar ones...
00:25:20.100 --> 00:25:33.300
We have x⁶ times y¹ (which I will just leave as y), times z², times z^-3, all over x⁴.
00:25:33.300 --> 00:25:35.800
And that is all we have on the bottom at this point.
00:25:35.800 --> 00:25:39.200
So, we see that we have z², and we see we have z^-3; so that will cancel out,
00:25:39.200 --> 00:25:50.300
and we will get -1, because -3 + 2...z² times z^-3 combine through addition,
00:25:50.300 --> 00:25:55.600
because they are both just multiplying each other, so we have z^2 - 3,
00:25:55.600 --> 00:26:00.600
which becomes z^-1, which is how we get what we have right here.
00:26:00.600 --> 00:26:11.900
x⁶ divided by x⁴ will cancel out all but two of these, which we could also see as x^6 - 4, which equals x².
00:26:11.900 --> 00:26:16.900
So, we have x² times y times z^-1.
00:26:16.900 --> 00:26:24.700
And since z^-1 is 1/z, we can write this as x²y, all over z.
00:26:24.700 --> 00:26:29.100
Once again, like our first example, there are other things that we could have done at various points.
00:26:29.100 --> 00:26:36.300
If we wanted to, we could have broken off here, and we probably could have written this as z²/x⁴,
00:26:36.300 --> 00:26:45.800
on our next step, times x⁶y⁹, and then z^-3/y⁸.
00:26:45.800 --> 00:26:51.900
At this point, we see z^-3, so we could move that over, and we could do x⁶/x⁴,
00:26:51.900 --> 00:26:56.500
times y⁹/y⁸, times z²...
00:26:56.500 --> 00:27:00.400
and since we had z^-3, we could also write that as z³.
00:27:00.400 --> 00:27:10.100
At this point, we have x^6 - 4 times y^9 - 8 times z^2 - 3.
00:27:10.100 --> 00:27:19.200
So, we have x² times y¹ times z^-1, which also becomes x²y/z.
00:27:19.200 --> 00:27:24.300
Or, if we wanted to, we could also just say, "We have 9 y's on top and 8 y's on the bottom;
00:27:24.300 --> 00:27:30.200
so all of them will cancel on the bottom, and one will be left on the top," and similar things with the x's and the z's.
00:27:30.200 --> 00:27:33.200
So, there are a variety of ways to look at these things, once you get into this.
00:27:33.200 --> 00:27:36.700
And once again, it is about developing an intuition and just doing it a bunch of times.
00:27:36.700 --> 00:27:45.000
And also, just be comfortable in the fact that whatever way you choose, as long as you follow the rules, they will all end up working out eventually.
00:27:45.000 --> 00:27:53.700
The third example: Simplify ^n√(5n(5^3n))^2n, divided by 5^6n².
00:27:53.700 --> 00:27:56.500
All right, this is a great one to show two different ways to approach this.
00:27:56.500 --> 00:28:00.300
Let's leave that nth root intact in our first one.
00:28:00.300 --> 00:28:02.700
nth root of 5^n times 5^3n:
00:28:02.700 --> 00:28:12.300
well, 5^n times 5^3n, because they are multiplying, will go through addition: 5^n + 3n.
00:28:12.300 --> 00:28:20.300
So, we have 5^4n, all raised to the 2n, over, still, 5^6n².
00:28:20.300 --> 00:28:23.000
Let's expand that radical a bit, so we see the whole thing.
00:28:23.000 --> 00:28:30.900
It equals...it still has that nth root, ^n√(5^4n(2n),
00:28:30.900 --> 00:28:39.400
because it was exponentiation on exponentiation, 4n on 2n, over 5^6n²),
00:28:39.400 --> 00:28:50.600
equals ^n√(5⁸) (4 times 2 is 8; n times n is n²) over 5^6n².
00:28:50.600 --> 00:28:54.900
Now, at this point, you might be tempted to cancel out our n²'s, but that would be improper,
00:28:54.900 --> 00:29:00.700
because it is not like canceling 3/3, where we can cancel both of them, because they are dividing.
00:29:00.700 --> 00:29:03.000
It is different, because it is about how many times they show up.
00:29:03.000 --> 00:29:10.500
So, we have to use the rule that we have, which is that, when we have a fraction with something on top and something on the bottom,
00:29:10.500 --> 00:29:13.400
it subtracts if they have the same thing on the bottom.
00:29:13.400 --> 00:29:16.600
They are both 5's, so they can use this rule.
00:29:16.600 --> 00:29:24.900
It is still the nth root, so it is going to be 5^8n² - 6n².
00:29:24.900 --> 00:29:36.200
So, those are common terms, so that becomes...8n² - 6n²...8 minus 6 is 2, but it still has n², so it is 5^2n².
00:29:36.200 --> 00:29:51.100
nth root is just the same thing as saying 1/n; so it is 5^2n²(1/n) = 5^2n.
00:29:51.100 --> 00:29:59.400
Great; an alternative way we could have done this, though, is that taking something to the nth root is 1/n, whenever we do it.
00:29:59.400 --> 00:30:05.100
So, 5^n times 5^3n...once again, that will be 5^4n, because they add together.
00:30:05.100 --> 00:30:09.300
And that is all raised to the 2n, over 5^6n².
00:30:09.300 --> 00:30:17.100
Now, it was a radical of the whole thing, so it has to be that the whole thing is raised to the 1/n.
00:30:17.100 --> 00:30:23.600
Now, we can distribute this, and we can say that that is 5^4n.
00:30:23.600 --> 00:30:33.200
Let's leave that as 2n times 1/n, over 5^6n² times 1/n,
00:30:33.200 --> 00:30:38.300
because this 1/n will get applied to the top and to the bottom of our fraction.
00:30:38.300 --> 00:30:54.000
5^4n...well, these n's cancel out; this 1/n cancels out the squared and leaves the n, so we have (5^4n)²/5^6n.
00:30:54.000 --> 00:31:01.400
5^4n(2), because it was exponentiation on exponentiation, and still 5^6n,
00:31:01.400 --> 00:31:15.000
equals 5^8n/5^6n, equals 5^8n - 6n, which equals 5^2n, just as well.
00:31:15.000 --> 00:31:19.400
Great; so, these are two different ways of doing it, very different approaches--going through the inside,
00:31:19.400 --> 00:31:25.000
or going from the outside in or inside out--but they both end up giving us the exact same answer.
00:31:25.000 --> 00:31:27.700
One of the great things about all of these rules is that they all work together.
00:31:27.700 --> 00:31:30.200
There is no preference of one rule versus the other.
00:31:30.200 --> 00:31:33.400
So, sometimes it generates various different paths that we could go.
00:31:33.400 --> 00:31:38.300
But you will develop an intuition; and once again, they all will end up working out.
00:31:38.300 --> 00:31:41.900
Just make sure you practice these things on your own, and you will develop a sense for how this works.
00:31:41.900 --> 00:31:44.200
And it will get faster and faster, the more you practice it.
00:31:44.200 --> 00:31:53.400
All right, another example: f(x) = 7x^2/3 - 2; g(x) = x^6/5; give f composed with g(x), and simplify.
00:31:53.400 --> 00:31:56.900
Now, remember: the first thing, when we talked about function composition:
00:31:56.900 --> 00:32:05.000
f(g(x)) is almost always the way we want to switch to writing this, f acting on g acting on x.
00:32:05.000 --> 00:32:13.500
What is g(x)? g(x) is x^6/5, so it is f(x^6/5).
00:32:13.500 --> 00:32:21.600
So now, it is not about the x; it is about where the box of input goes into our formula for that function.
00:32:21.600 --> 00:32:26.100
So, f(box) = 7(box)^2/3 - 2.
00:32:26.100 --> 00:32:35.800
In this case, our box is x^6/5; we have 7 times box, x^6/5;
00:32:35.800 --> 00:32:42.000
and then, that box raised to the 2/3 - 2, because that is what our whole function said before.
00:32:42.000 --> 00:32:49.300
So, it is 7 times x^6/5; because it is exponentiation on exponentiation, it is multiplication;
00:32:49.300 --> 00:33:00.600
2/3 - 2; 7 times x; 6/5 times 2/3; we notice that 6 can be broken into 3 times 2, so that knocks out this 3 and this 3.
00:33:00.600 --> 00:33:08.100
And we are left with 2 times 2 on the top, so that is 4/5, minus 2.
00:33:08.100 --> 00:33:15.300
So, 7 times x^4/5 - 2 is what f composed with g comes out to be.
00:33:15.300 --> 00:33:21.900
The final example: let x be a number such that 7^x equals 3/5; what is 49^x?
00:33:21.900 --> 00:33:26.100
Now, at first, you might see a problem like this, and it is completely confusing, because you have no idea what to do.
00:33:26.100 --> 00:33:31.800
But notice: we have 7 here; we have 49 here; so there is going to be some sort of clever trick
00:33:31.800 --> 00:33:35.700
connecting the fact that 49 has something to do with 7.
00:33:35.700 --> 00:33:43.200
How is 49 related to 7? 49 is equal to 7².
00:33:43.200 --> 00:33:47.500
So, there is this connection between 49 and 7, so we can use that.
00:33:47.500 --> 00:33:58.200
We can now apply that, and we can say, "49^x...we know that 49 is equal to 7², so we can just substitute that out."
00:33:58.200 --> 00:34:05.400
So, put 49^x in parentheses; it is the same thing as (7²)^x.
00:34:05.400 --> 00:34:08.000
There is nothing you can do to just stop substitution like that.
00:34:08.000 --> 00:34:16.900
So, (7²)^x--that means 7 to the 2 times x, but we could also write that as 7 to the x times 2,
00:34:16.900 --> 00:34:22.500
which we could then write as 7^x, all raised to the 2, which would be...
00:34:22.500 --> 00:34:26.300
we know what 7^x is--it is 3/5!
00:34:26.300 --> 00:34:38.000
So, we have 3/5, all raised to the 2, which means we have 9/25, because we square the 3, and we square the 5.
00:34:38.000 --> 00:34:43.400
3/5 squared means that the square will go onto the 3 and go onto the 5.
00:34:43.400 --> 00:34:46.600
All right, exponents are pretty cool stuff; they are really, really powerful.
00:34:46.600 --> 00:34:49.300
It is important to get a good grasp of just working with them, though.
00:34:49.300 --> 00:34:52.400
The only way that you will be able to get really comfortable with them is doing some practice.
00:34:52.400 --> 00:34:57.100
So, just make sure that you do some practice with exponentiation, using exponents of various types.
00:34:57.100 --> 00:35:01.500
But once you get in a bit of practice, you will get used to it; they are skills that stick with you.
00:35:01.500 --> 00:35:09.400
And as long as you stay with this x^a times x^b = x^a + b,
00:35:09.400 --> 00:35:14.800
as long as you stay with that idea, you can figure out everything else if you get in a situation where you forget one of the rules.
00:35:14.800 --> 00:35:17.000
All right, we will see you at Educator.com later--goodbye!