Introduction to Exponents

>> Professor Perez: Hey!
This is Professor Perez again.
Today, we're going to work on an introduction to exponents.
Of course, we can't get started without Charlie.
He better be ready to go!
Charlie! What are you doing over there, what are you on vacation or something?
Take out a piece of paper and a pencil and get ready to go!
We've got to get started here.
Today we're doing introduction to exponents.
Okay, let's start with a review right here.
5 times 2, Charlie.
What does that mean?
>> Charlie: 5 plus 5.
>> Professor Perez: Very nice.
5 plus 5. Which is equal to what?
>> Charlie: 10.
>> Professor Perez: 10.
Okay, now, watch this Charlie.
Here we have a 5 with a little 2 up there, right?
Well, in this situation, the 5 is considered to be the base.
And the 2 up there Charlie, is what we call the exponent.
That's what we're going to be talking about.
What does that exponent mean.
Well first of all, how do we say this?
You can say it as 5 raised to the second power, Charlie.
But 5 raised to the second power can be said differently because a number raised
to the second power occurs quite often, and we have a shortcut name, we call that 5 squared.
So you can say it either way.
5 raised to the second power, or simply, 5 squared.
Well, what we want to know now, is what does it mean?
Okay, Charlie?
Remember, 5 times 2 meant 5 plus 5.
5 raised to the second power means 5 times 5.
You see the difference?
This one's 5 plus 5, that's 5 times 5.
And what is 5 times 5, Charlie?
>> Charlie: 25.
>> Professor Perez: 25, very nice, here, okay, so here we go.
We have 2 times 4, and over here, we'll put 2 raised to the fourth power.
Now Charlie, what does 2 times 4 actually mean?
>> Charlie: 2 plus 2 plus 2 plus 2.
>> Professor Perez: 2 plus 2 plus 2 plus 2.
Very nice.
Which is what?
>> Charlie: 8.
>> Professor Perez: 8, okay.
Now 2 raised to the fourth power, Charlie, what does that mean?
>> Charlie: 2 times 2 times 2 times 2.
>> Professor Perez: That's right.
2 times 2 times 2 times 2.
Okay, now, what does 2 times 2 times 2 times 2 equal, Charlie?
>> Charlie: 16?
>> Professor Perez: 16.
Very nice.
So, 2 times 4 is 8.
2 raised to the fourth ends up being 16.
Okay, let's do another one.
3 times 2 and over here, we'll do 3 raised to the second power, or 3 squared.
I'll call it 3 squared now.
3 times 2.
What does it mean, Charlie?
>> Charlie: 3 plus 3.
>> Professor Perez: 3 plus 3 which is?
>> Charlie: 6.
>> Professor Perez: 6.
Now what does 3 squared mean, Charlie?
>> Charlie: 3 times 3.
>> Professor Perez: 3 times 3, that's very nice.
And what does that equal?
>> Charlie: 9.
>> Professor Perez: 9.
Very nice there Charlie.
Okay, now, what we're going to do is look at this one.
Don't get scared!
It's x raised to the fifth power, Charlie.
Just use the force, Charlie.
What does it mean?
>> Charlie: x times x times x times x times x.
>> Professor Perez: That's right.
This means x times x times x times x times x. That's what it is, okay?
That's what it represents.
Now, let's look at 3 to the fourth power, Charlie.
What does it mean?
>> Charlie: 3 times 3 times 3 times 3.
>> Professor Perez: 3 times 3 times 3 times 3.
Now, how do we calculate 3 times 3 times 3 times 3?
Well, some people just go 3 times 3 is 9, times 3 is 27, and 27 times 3...
well then you get stuck.
But we're going to bust out some Kung-Fu, watch.
Remember, when numbers are being all multiplied together,
you can group them or multiply in any order.
Here we're going to group them.
We're going to bust out some Kung-Fu.
Watch. What's 3 times 3, Charlie?
>> Charlie: 9.
>> Professor Perez: That's 9.
3 times 3 is?
>> Charlie: 9.
>> Professor Perez: Now what's 9 times 9?
>> Charlie: 81.
>> Professor Perez: 81 and that's it.
See some people remember 9 times 9 is 81, that's right.
The answer is 81.
Very nice there, Charlie!
Let's move on.
Write that as a word statement.
In other words, how do you say that...well I don't want
to tell you the answer, that 2 with a little 3 up there.
How do you say that?
That is what?
>> Charlie: 2?
>> Professor Perez: Okay...
>> Charlie: ...raised to the third?
>> Professor Perez: Raised to the third power, very nice there Charlie.
That's 2 raised to the third power.
But it has a shortcut name.
Right? How could you say it?
>> Charlie: 2 cubed?
>> Professor Perez: 2 cubed, very nice there.
Okay, now let's go to this next one there.
Write that one as a word statement Charlie, go ahead and say it.
>> Charlie: 6 raised to the seventh.
>> Professor Perez: Very nice there Charlie, 6 raised to the seventh power.
Very nice, okay.
Now, what we're going to do now, is we're going to have to look at what does it mean
for an exponent to be equal to a zero.
That's a tough one.
We're in pre-algebra, and we're going to look at this from a pattern standpoint.
You'll see what I mean, watch.
2 to the fourth, what does that mean, Charlie?
>> Charlie: 2 times 2 times 2 times 2.
>> Professor Perez: Okay, which is equal to 16.
Now, 2 cubed means 2 times 2 times 2 which is 8.
2 to the f...or 2 squared, sorry, means 2 times 2 which is 4.
2 to the first power means you only have one 2, it's 2.
Okay? Now the question is, what is 2 to the 0 power?
Well, we have to look at a pattern.
Now notice over there, Charlie, with those numbers.
We have a 16, an 8, a 4, and a 2.
What's going on?
>> Charlie: Cut in half?
>> Professor Perez: They end up being cut in half.
So if we continue the pattern, what's half of 16?
>> Charlie: 8
>> Professor Perez: 8 and what's half of 8?
>> Charlie: 4.
>> Professor Perez: 4.
And what's half of 4?
>> Charlie: 2.
>> Professor Perez: 2.
Well what's half of 2?
>> Charlie: 1.
>> Professor Perez: It is 1.
So basically what we're going is we're dividing by 2.
Simply because 2 is our base.
We're losing a 2 every time.
Okay? 16 divided 8, I'm sorry, 16 divided by 2 is 8.
8 divided by 2 is 4.
4 divided by 2 is 2.
And 2 divided by 2 is 1.
And notice, with our pattern, we're seeing, hey, 2 to the 0 is equal to 1!
Hmmm! Okay?
Let's try it with another base.
Let's try it with a 3 now.
3 to the fourth means 3 times 3 times 3 times 3 which we just saw previously was 81.
3 to the third, or 3 cubed means 3 times 3 times 3 which is 27.
3 squared is 3 times 3 which is 9.
3 to the first power is just a 3 which is 3.
Now again, what is 3 to the 0?
Well, it we look at our pattern here, we have 81, 27, 9, and 3.
Well, I'll help you out here, 81 divided by 3 is 27.
27 divided by 3 is 9.
9 divided by 3 is 3.
And if I continue the pattern here, we have base 3, 3 divided by 3 is 1,
and so again, we see 3 to the 0 is 1.
So here we've done it with two different bases, and again, look at the pattern.
We've got a result of 1 whenever we raised to the 0 power.
So if we do it with 4, I'll go very quickly with this one.
We have 4 to the fourth, 4 cubed, 4 squared, 4 to the first, and again what's 4 to the 0.
Again, if we take a look at that pattern here, we are dividing by 4's, okay,
because every time we're losing a 4.
And so 256 divided by 4 is 64 divided by 4 is 16,
16 divided by 4 is 4, and 4 divided by 4 is a 1.
And so if you do this with any number, any base, a positive or a negative,
you will always get 1 when you raise to the 0 power.
So our conclusion is, any number raised to the 0 power except 0, is equal to 1.
It will not work with 0, because notice here we're dividing by 2's, here we're dividing
by 3's, here we're dividing by 4's, we cannot divide by 0.
So, 0 to the 0 is undefined.
That's a special case there, so, any number raised to the 0 power except 0 is equal to 1.
So, let's look at some other problems here.
2 squared times 2 cubed.
2 squared means 2 times 2, and 2 raised to the third power means 2 times 2 times 2,
so how do we write five 2's being to each other, Charlie?
>> Charlie: 2 to the fifth!
>> Professor Perez: That's 2 to the fifth, very nice, okay.
Now, what is 2 to the fifth?
2 times 2 is 4, times 2 is 8 times 2 is 16...
what'd you get, Charlie?
>> Charlie: 32.
>> Professor Perez: 32, very nice.
Let's throw in some variables, x squared times x cubed.
What does x squared mean, Charlie?
>> Charlie: x times x.
>> Professor Perez: x times x, what's x cubed?
>> Charlie: x times x times x.
>> Professor Perez: x times x times x,
now how do you write five x's being multiplied to each other?
>> Charlie: x to the fifth.
>> Professor Perez: x raised to the fifth, that's it.
Now, don't get confused with this.
This is 2 x plus 3 x. If you have 2 apples, and somebody gives you 3 more apples,
how many apples do you have, Charlie?
>> Charlie: 5 apples.
>> Professor Perez: 5 apples, that's right.
So if you have 2 x's and somebody gives you 3 more x's, how many x's do you have Charlie?
>> Charlie: 5 x's.
>> Professor Perez: 5 x's, that' right, because 2x means x plus x,
3x means x plus x plus x, which is 5 x's.
5 x's being added together is 5 times x. Anyway, that's it for now.
We'll see you again soon!