We’d express this really small number in scientific notation.
Let’s just see how small it is.
It is one, two, three, four, five, six, seven, eight, nine zeroes.
And then a 3457.
All of that is behind the decimal point.
Now this is a bit of review about scientific notation.
It means expressing a number in a form– let me give an example, it could be 5 times 10 to the fifth power.
This would be an example of writing something in scientific notation.
And when you write something in scientific notation, the number that you’re multiplying by a power of ten, that number right there has to be greater than or equal to 1, and it should be less than 10.
And of course, this right here is going to be a power of 10.
So for example, if I were to write 5 times 9 to the fifth, not scientific notation.
If I were to write 50 times 10 to the fifth, not quite scientific notation.
Sometimes this kind of gets by for scientific notation while you’re doing computation.
But if you really wanted to write this in scientific notation, you would write this as 5 times 10, which is 50 times 10 to the fifth.
And 10 times 10 to the fifth is 10 to the sixth.
So you’d write this as 5 times 10 to the sixth.
So your goal is always for this number right here to be greater than or equal to 1, less than 10.
Likewise, if I wrote 0.9 times 10 squared, once again, not in scientific notation.
You would want to write 0.9 as 9 times 10 to the negative 1 times 10 squared.
And then this becomes 9 times 10.
Which makes sense, because 0.9 times 100 is 90.
9 times 10 is 100.
We could say 9 times 10 to the first power.
So that was a bit of review.
Let’s do it for this problem right here.
I guess a quick way to think about converting this to scientific notation, think about how many leading zeroes there are including the first non zero number right there.
So there are 10 digits to the right if you include this 3.
You have nine zeroes and then you have this 3.
So a very quick way to do it, and I’ll explain in a second why it works, is we could rewrite this right here as 3.457 times 10 to the, we had nine zeroes, and then I count the 3, 10.
So times 10 to the negative 10 power.
And the reason why this makes sense is, when you multiply something to a negative power, you shift the decimal over to the left.
Let me write it this way.
Start with 3.457.
And when you multiply it times 10 to the negative 10, you’re going to shift the decimal to the left 10 spaces.
So if you shift the decimal to the left 10 spaces, what do you get?
So if you shift it once, twice, three, four, five, six, seven, eight, nine, ten.
And of course, you can’t just leave open space here.
There’s going to be zeroes there when you shift it.
So you’re going to have one, two, three, four, five, six, seven, eight, nine zeroes.
And you can put a zero out here just to make sure that you understand that there is a zero in front of that decimal.
And you get exactly what we got up there.
Another way to think about it is, if you wanted to start with this really small number and you wanted to get to 3.457, you could multiply it by 10 to the tenth.
That would shift this decimal 10 spaces to the right.
So you could do this.
And these are all the multiple ways of looking at the exact same thing.
You could write that 3.457 is equal to this character.
Or I could even copy and paste this character. Times 10 to the tenth power.
If you multiply by a positive exponent, every time you multiply by 10 it shifts the decimal to the right.
It makes the number bigger.
So if you multiply this by 10 to the tenth, this guy is going to move one, two, three, four, five, six, seven, eight, nine, ten.
It would shift it right there.
And you’d get 3.457.
So that’s the whole idea.
3.457 is this number times 10 to the tenth.
So if you wanted to solve for this number, you would multiply both sides of this equation by 10 to the negative 10.
So times 10 to the negative 10, or that’s the equivalent of dividing by 10 to the tenth.
These cancel out.
Negative tenth is 10 to the zero or just 1.
And you get this thing over here being equal to 3.457 times 10 to the negative 10.
I just wrote it in a different order over here.
>>Matt Cutts: Today’s question comes from SurfVoucher in Costa Rica.
The question is, “How come the search queries report in Webmaster Tools shows some keywords ranking and when we search for them they are nowhere to be found?
” Well, there’s a few reasons why this might happen.
First off, maybe you’ve got personalized search and maybe you’ve clicked on things or not clicked on things and so you’re getting different search results than a lot of other people might be seeing.
The second thing is it can depend on the country.
So maybe the query bank returns you number one in Australia but not number one in America or-or the United States, something like that.
So we-we can even personalize down to a metro level.
So you might rank well in Chicago but not rank all that well in Orlando or Miami Beach.
So the other thing to think about is that the rankings do change over time.
So you might be hitting a different data center than some other places.
And you might have ranked for awhile and then not be ranking now, or you might be ranking higher and you didn’t rank for awhile.
So be aware that algorithms do change and the rankings do change as a result, and then as we’re pushing data that can result in different rankings as well.
So those are at least three reasons why you mot not, might not be seeing yourself, but let me give you a little bit of an optimistic reason to pay attention.
The fact that the Search Queries Report in the Webmaster console is showing that you ranked somewhere in the world for this is a really good sign.
In fact, if you don’t show up what that means is there’s a keyword that you have ranked for that you don’t currently see yourself for.
So that’s sort of a keyword you might wanna put a little more attention into and try to optimize so that you show up on the first page or number one or whatever for more places — for more personalized search results — that sort of thing.
So in essence, whenever you see something showing up in the Search Queries Report and you don’t see it whenever you do that search, that might be a re, a reason to be very happy because you’ve now identified a keyword which if you put a little bit of work into it you might be able to help have that query rank you just a little bit higher to where everyone will see it and not just some people.
So don’t just get discouraged.
There always can be data differences and-and reasons why you might or might not see it, but if you keep workin’ you might be able to see it a lot more often which will also mean that a lot more people see it a lot more often.
What I want to do in this video is build up some tools in our tool kit for dealing with sums and differences of random variables.
So let’s say that we have two random variables, x and y, and they are completely independent.
They are independent random variables.
And I’m just going to go over a little bit of a notation here.
If we wanted to know the expected, or if we talked about the expected value of this random variable x, that is the same thing as the mean value of this random variable x.
If we talk about the expected the value of y, that is the same thing as the mean of y.
If we talk about the variance of the random variable x, that is it the same thing as the expected value of the squared distances between our random variable x and its mean.
And that right there squared.
So the expected value of these squared differences, and that you could also use the notation sigma squared for the random variable x.
This is just a review of things we already know, but I just want to reintroduce it because I’ll use this to build up some of our tools.
So you do the same thing with this with random variable y.
The variance of random variable y is the expected value of the squared difference between our random variable y and the mean of y, or the expected value of y, squared.
And that’s the same thing as sigma squared of y.
There is the variance of y.
Now you may or may not already know these properties of expected values and variances, but I will reintroduce them to you.
And I won’t go into some rigorous proof– actually, I think they’re fairly easy to digest.
So one is is that if I have some third random variable, let’s say I have some third random variable that is defined as being the random variable x plus the random variable y.
Let me stay with my colors just so everything becomes clear.
The random variable x plus the random variable y.
What is the expected value of z going to be?
The expected the value of z is going to be equal to the expected value of x plus y.
And this is a property of expected values– I’m not going to prove it rigorously right here– but the expected value of x plus the expected value of y, or another way to think about this is that the mean of z is going to be the mean of x plus the mean of y.
Or another way to view it is if I wanted to take, let’s say I have some other random variable.
I’m running out of letters here.
Let’s say I have the random variable a, and I define random variable a to be x minus y.
So what’s its expected value going to be?
The expected value of a is going to be equal to the expected value of x minus y, which is equal to– you could either view it as the expected value of x plus the expected value of negative y, or the expected value of x minus the expected value of y, which is the same thing as the mean of x minus the mean of y.
So this is what the mean of our random variable a would be equal to.
And all of this is review and I’m going to use this when we start talking about the distributions that are sums and differences of other distributions.
Now let’s think about what the variance of random variable z is and what the variance of random variable a is.
So the variance of z– and just to kind of always focus back on the intuition, it makes sense.
If x is completely independent of y and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two because they are unrelated.
If my expected value here is 5 and my expected value here is 7, completely reasonable that my expected value here is 12, assuming that they’re completely independent.
Now if we have a situation, so what is the variance of my random variable z?
And once again, I’m not going do a rigorous proof here, this is really just a property of variances.
But I’m going to use this to establish what the variance of our random variable a is.
So if this squared distance on average is some variance, and this one is completely independent, it’s squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances.
So this is going to be equal to the variance of random variable x plus the variance of random variable y.
Or another way of thinking about it is that the variance of z, which is the same thing as the variance of x plus y, is equal to the variance of x plus the variance of random variable y.
Hopefully that make some sense.
I’m not proving it to you rigorously.
And you’ll see this in a lot of statistics books.
Now what I want to show you is that the variance of random variable a is actually this exact same thing.
And that’s the interesting thing, because you might say, hey, why wouldn’t it be the difference?
We had the differences over here.
So let’s experiment with this a little bit.
The variance– so I’ll just write this– the variance of random variable a is the same thing as the variance of– I’ll write it like this– as x minus y, which is equal to– you could view it this way– which is equal to the variance of x plus negative y.
These are equivalent statements.
So you could view this as being equal to– just using this over here, the sum of these two variances, so it’s going to be equal to the sum of the variance of x plus the variance of negative y.
Now what I need to show you is that the variance of negative y, of the negative of that random variables are going to be the same thing as the variance of y.
So what is the variance of negative y?
The variance of negative y is the same thing as the variance of negative y, which is equal to the expected value of the distance between negative y and the expected value of negative y squared.
That’s all the variance actually is.
Now what is the expected value of negative y right over here?
Actually, even better let me factor out a negative 1.
So what’s in the parentheses right here, this is the exact same thing as negative 1 squared times y plus the expected value of negative y.
So that’s the same exact same thing in the parentheses, squared.
So everything in magenta is everything in magenta here, and it is the expected value of that thing.
Now what is the expected value of negative y?
The expected value of negative y– I’ll do it over here– the expected value of the negative of a random variable is just a negative of the expected value of that random variable.
So if you look at this we can re-write this– I’ll give myself a little bit more space– we can re-write this as the expected value– the variance of negative y is the expected value– this is just 1.
Negative 1 squared is just 1.
And over here you have y, and instead just write plus the expected value of negative y, that’s the same thing as minus the expected value of y.
So you have that, and then all of that squared.
Now notice, this is the exact same thing by definition as the variance of y.
So what we just showed you just now, so this is the variance of y.
So we just showed you is that the variance of the difference of two independent random variables is equal to the sum of the variances.
You could definitely believe this, it’s equal to the sum of the variance of the first one plus the variance of the negative of the second one.
And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense.
Your distance from the mean is going to be– it doesn’t matter whether you’re taking the positive or the negative of the variable.
You just cared about absolute distance.
So it makes complete sense that that quantity and that quantity is going to be the same thing.
Now the whole reason why I went through this exercise, kind of the important takeaways here is that the mean of differences right over here– so I could re-write it as the differences of the random variable is the same thing as the differences of their means.
And then the other important takeaway, and I’m going to build on this in the next few videos, is that the variance of the difference– if I define a new random variable is the difference of two other random variables, the variance of that random variable is actually the sum of the variances of the two random variables.
So these are the two important takeaways that we’ll use to build on in future videos.
Anyway, hopefully that wasn’t too confusing.
If it was, you can kind of just accept these at face value and just assume that these are tools that you can use.
In this video, I want to do a bunch of examples involving exponent properties.
But, before I even do that, let’s have a little bit of a review of what an exponent even is.
So let’s say I had 2 to the third power.
You might be tempted to say, oh is that 6?
And I would say no, it is not 6.
This means 2 times itself, three times.
So this is going to be equal to 2 times 2 times 2, which is equal to 2 times 2 is 4.
4 times 2 is equal to 8.
If I were to ask you what 3 to the second power is, or 3 squared, this is equal to 3 times itself two times.
This is equal to 3 times 3.
Which is equal to 9.
Let’s do one more of these.
I think you’re getting the general sense, if you’ve never seen these before.
Let’s say I have 5 to the seventh power.
That’s equal to 5 times itself, seven times.
5 times 5 times 5 times 5 times 5 times 5 times 5.
That’s seven, right?
One, two, three, four, five, six, seven.
This is going to be a really, really, really, really, large number and I’m not going to calculate it right now.
If you want to do it by hand, feel free to do so.
Or use a calculator, but this is a really, really, really, large number.
So one thing that you might appreciate very quickly is that exponents increase very rapidly.
5 to the 17th would be even a way, way more massive number.
But anyway, that’s a review of exponents.
Let’s get a little bit steeped in algebra, using exponents.
So what would 3x– let me do this in a different color– what would 3x times 3x times 3x be?
Well, one thing you need to remember about multiplication is, it doesn’t matter what order you do the multiplication in.
So this is going to be the same thing as 3 times 3 times 3 times x times x times x.
And just based on what we reviewed just here, that part right there, 3 times 3, three times, that’s 3 to the third power.
And this right here, x times itself three times.
that’s x to the third power.
So this whole thing can be rewritten as 3 to the third times x to the third.
Or if you know what 3 to the third is, this is 9 times 3, which is 27.
This is 27 x to the third power.
Now you might have said, hey, wasn’t 3x times 3x times 3x.
Wasn’t that 3x to the third power?
Right?
You’re multiplying 3x times itself three times.
And I would say, yes it is.
So this, right here, you could interpret that as 3x to the third power.
And just like that, we stumbled on one of our exponent properties.
Notice this.
When I have something times something, and the whole thing is to the third power, that equals each of those things to the third power times each other.
So 3x to the third is the same thing is 3 to the third times x to the third, which is 27 to the third power.
Let’s do a couple more examples.
What if I were to ask you what 6 to the third times 6 to the sixth power is?
And this is going to be a really huge number, but I want to write it as a power of 6.
Let me write the 6 to the sixth in a different color.
6 to the third times 6 to the sixth power, what is this going to be equal to?
Well, 6 to the third, we know that’s 6 times itself three times.
So it’s 6 times 6 times 6.
And then that’s going to be times– the times here is in green, so I’ll do it in green.
Maybe I’ll make both of them in orange.
That is going to be times 6 to the sixth power.
Well, what’s 6 to the sixth power?
That’s 6 times itself six times.
So, it’s 6 times 6 times 6 times 6 times 6.
Then you get one more, times 6.
So what is this whole number going to be?
Well, this whole thing– we’re multiplying 6 times itself– how many times?
One, two, three, four, five, six, seven, eight, nine times, right?
Three times here and then another six times here.
So we’re multiplying 6 times itself nine times.
3 plus 6.
So this is equal to 6 to the 3 plus 6 power or 6 to the ninth power.
And just like that, we/ve stumbled on another exponent property.
When we take exponents, in this case, 6 to the third, the number 6 is the base.
We’re taking the base to the exponent of 3.
When you have the same base, and you’re multiplying two exponents with the same base, you can add the exponents.
Let me do several more examples of this.
Let’s do it in magenta.
Let’s say I had 2 squared times 2 to the fourth times 2 to the sixth.
Well, I have the same base in all of these, so I can add the exponents.
This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power.
And hopefully that makes sense, because this is going to be 2 times itself two times, 2 times itself four times, 2 times itself six times.
When you multiply them all out, it’s going to be 2 times itself, 12 times or 2 to the 12th power.
Let’s do it in a little bit more abstract way, using some variables, but it’s the same exact idea.
What is x to the squared or x squared times x to the fourth?
Well, we could use the property we just learned.
We have the exact same base, x.
So it’s going to be x to the 2 plus 4 power.
It’s going to be x to the sixth power.
And if you don’t believe me, what is x squared?
x squared is equal to x times x.
And if you were going to multiply that times x to the fourth, you’re multiplying it by x times itself four times.
x times x times x times x.
So how many times are you now multiplying x by itself?
Well, one, two, three, four, five, six times.
x to the sixth power.
Let’s do another one of these.
The more examples you see, I figure, the better.
So let’s do the other property, just to mix and match it.
Let’s say I have a to the third to the fourth power.
So I’ll tell you the property here, and I’ll show you why it makes sense.
When you add something to an exponent, and then you raise that to an exponent, you can multiply the exponents.
So this is going to be a to the 3 times 4 power or a to the 12th power.
And why does that make sense?
Well this right here is a to the third times itself four times.
So this is equal to a to the third times a to the third times a to the third times a to the third.
Well, we have the same base, so we can add the exponents.
So there’s going to be a to the 3 times 4, right?
This is equal to a to the 3 plus 33 plus three plus 3 power, which is the same thing is a the 3 times 4 power or a to the 12th power.
So just to review the properties we’ve learned so far in this video, besides just a review of what an exponent is, if I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power.
We saw that right here.
x squared times x to the fourth is equal to x to the sixth, 2 plus 4.
We also saw that if I have x times y to the a power, this is the same thing is x to the a power times y to the a power.
We saw that early on in this video.
We saw that over here.
3x to the third is the same thing as 3 to the third times x to the third.
That’s what this is saying right here.
3x to the third is the same thing is 3 to the third times x to the third.
And then the last property, which we just stumbled upon is, if you have x to the a and then you raise that to the bth power, that’s equal to x to the a times b.
And we saw that right there.
a to the third and then raise that to the fourth power is the same thing is a to the 3 times 4 or a to the 12th power.
So let’s use these properties to do a handful of more complex problems.
Let’s say we have 2xy squared times negative x squared y squared times three x squared y squared.
And we wanted to simplify this.
This you can view as negative 1 times x squared times y squared.
So if we take this whole thing to the squared power, this is like raising each of these to the second power.
So this part right here could be simplified as negative 1 squared times x squared squared, times y squared.
And then if we were to simplify that, negative 1 squared is just 1, x squared squared– remember you can just multiply the exponents– so that’s going to be x to the fourth y squared.
That’s what this middle part simplifies to.
And let’s see if we can merge it with the other parts.
The other parts, just to remember, were 2 xy squared, and then 3x squared y squared.
Well now we’re just going ahead and just straight up multiplying everything.
And we learned in multiplication that it doesn’t matter which order you multiply things in.
So I can just rearrange.
We’re just going and multiplying 2 times x times y squared times x to the fourth times y squared times 3 times x squared times y squared.
So I can rearrange this, and I will rearrange it so that it’s in a way that’s easy to simplify.
So I can multiply 2 times 3, and then I can worry about the x terms.
Let me do it in this color.
Then I have times x times x to the fourth times x squared.
And then I have to worry about the y terms, times y squared times another y squared times another y squared.
And now what are these equal to?
Well, 2 times 3.
You knew how to do that.
That’s equal to 6.
And what is x times x to the fourth times x squared.
Well, one thing to remember is x is the same thing as x to the first power.
Anything to the first power is just that number.
So you know, 2 to the first power is just 2.
3 to the first power is just 3.
So what is this going to be equal to?
This is going to be equal to– we have the same base, x.
We can add the exponents, x to the 1 plus 4 plus 2 power, and I’ll add it in the next step.
And then on the y’s, this is times y to the 2 plus 2 plus 2 power.
And what does that give us?
That gives us 6 x to the seventh power, y to the sixth power.
And I’ll just leave you with some thing that you might already know, but it’s pretty interesting.
And that’s the question of what happens when you take something to the zeroth power?
So if I say 7 to the zeroth power, What does that equal?
And I’ll tell you right now– and this might seem very counterintuitive– this is equal to 1, or 1 to the zeroth power is also equal to 1.
Anything that the zeroth power, any non-zero number to the zero power is going to be equal to 1.
And just to give you a little bit of intuition on why that is.
Think about it this way.
3 to the first power– let me write the powers– 3 to the first, second, third.
We’ll just do it the with the number 3.
So 3 to the first power is 3.
I think that makes sense.
3 to the second power is 9.
3 to the third power is 27.
And of course, we’re trying to figure out what should 3 to the zeroth power be?
Well, think about it.
Every time you decrement the exponent.
Every time you take the exponent down by 1, you are dividing by 3.
To go from 27 to 9, you divide by 3.
To go from 9 to 3, you divide by 3.
So to go from this exponent to that exponent, maybe we should divide by 3 again.
And that’s why, anything to the zeroth power, in this case, 3 to the zeroth power is 1.
See you in the next video.
My name is Marc Leny and I’m a just a geek hanging around the cyberspace just like you.
The purpose of this blog is to bring more knowledge to people around the world about business and everything, especially those information from videos from Youtube.
