Welcome to the presentation on exponents.
If I were to ask you what
-- let me make sure this pen is the right width--
if I were to ask you what two times three is,
well I think at this point this should be pretty easy for you.
That's the same thing as two plus two plus two, which equals six.
You didn't have to do this.
You all know that two times three is equal to six.
So what we're going to do now is
we're going to learn exponents which are the same thing
that multiplication is to addition, exponents are to multiplication.
I'll explain that in a second.
I know that probably confused you.
If I were to say what two to the third power is,
instead of two plus two plus two,
this is equal to two times two times two, which equals eight.
Or if I were to say three the second power,
that is equal to three times three.
Remember, I said three times two would be three plus three.
So the first three times three this equals nine,
and three plus three equals six.
The reason why I'm doing this is because there's always this temptation
when you first learn exponents is to multiply.
When I first learned it and I saw three to the second power or three squared,
I'd always be like oh, that's six.
But you also remember it's three times three,
which equals nine.
Let's do some more problems.
If I were to tell you negative four squared.
Once again, that's the same thing as
negative four times negative four.
Well, we all learn from multiplying negative numbers
that a negative times a negative is a positive.
And then four times four,
well this equals positive sixteen.
You don't have to write the positive,
I'm just doing that for emphasis.
If I would ask you what negative four to the third power is,
well that equals negative four times negative four
times negative four.
Well we know already that negative four times negative four,
that this equals sixteen, positive sixteen,
and then we multiply that times negative four.
And then that equals minus sixty-four.
So something very interesting here to observe.
When I took a negative number,
and we call this the base,
when the base is negative,
in this case, negative four,
and I raise it to an even power,
I got a positive number, right?
Negative four to an even power is positive sixteen,
and when I took a negative number to an odd power, to three,
I got a negative number.
And that makes sense because every time
you multiply by a negative number again,
it switches signs.
I'll show you the, I guess you'd call it
the simplest example.
Negative one to the one power is equal to negative one, right?
Because that's just negative one times itself one time.
And if I said negative one squared,
well that's negative one times negative one,
well that equals positive one.
But if I said negative one to the third power,
once again that's negative one times negative one times negative one.
Well now this equals negative one times negative one is positive one
times negative one equals negative one again.
So I could tell you
what negative one to the fifty-first power is.
Because fifty-one is odd,
we know that that is equal to negative one.
If it was a fifty then it would be a positive one.
Hope I didn't confuse you too much.
Let's do a couple more problems.
If I asked you what five to the third power is,
well that will go into five times five times five,
which equals one hundred and twenty-five.
So really if I were ask to ask
what negative five to the third power is,
that would be negative five times negative five times negative five,
which would be negative one hundred and twenty-five.
Now one principle of exponents
that might not seem completely intuitive to you at first
is when I raise something to the zero power.
So let's say I had two to the zero power.
It turns out that anything to the zero power is equal to one.
So two to the zero power is one,
three to the zero power is equal to one,
negative nine hundred to the zero power is equal to one.
And let me see if I can give you a little bit of intuition of
why that is actually the case.
So if I were to ask you,
let's do three to the fourth power.
That equals three times three times three times three,
which equals eighty-one.
three to the third power is equal to twenty-seven,
this three times three times three.
three to second power is equal to nine.
three to the first power is equal to three.
Now we're going to say what's three to zero power?
Well we already know, I already told you the rule,
anything with the zero power is equal to one,
but this will hopefully give you some intuition.
When we went from the fourth power to the third power,
we divided by three, right?
eighty-one divided by three is twenty-seven.
When you went from the third power to the second power,
we divided by three.
When we went from the second power from nine to three,
we divided by three.
So it kind of makes logical sense that
when we go from the first power to the zero power,
we'll just divide by three again.
So three divided by three is one.
Hopefully that gives you a little bit of an intuitive sense.
You might want to replay that and think about why that is.
And there's actually other aspects of exponents that
why this also makes sense,
why something to the zero power
is equal to one.
But let's just do some more problems in the time we have.
I don't want to get you too confused.
So if I were to ask you seven squared,
well that's seven times seven, that's forty-nine.
If I asked you negative six to the third power
-- parentheses around here so you know it's a whole negative six to the third power--
that equals negative six times negative six times negative six.
Negative six times negative six is positive thirty-six times negative six,
and that equals what?
It's one hundred and eighty and thirty-six,
that's minus two hundred and sixteen,
if my mental math is correct.
You could have actually multiplied it out.
I think you're getting the point at this point.
Oh, and another thing,
if I told you zero to the hundredth power,
well, that's pretty easy.
That's zero times itself one hundred times,
which is still equal to zero.
If I were to ask you one to the thousandth power,
well that's just equal to one, right?
You can multiply one by itself as many times as you want
and you're still going to get one.
And remember, if I had negative one to the one thousandth power,
well, this is an even exponent so you're still going to get one.
If it was negative one to the one thousand and one,
then it would be negative one.
I think you remember why this is,
because when you multiply a negative times itself an even number of times,
the negatives cancel out.
And then if you multiply it by a negative one more time
it becomes a negative number again.
Well let's just do some normal problems.
I just want to make sure you get the basics of exponents down.
If I were to tell you
-- let me think of a good one--
eight squared,
that equals eight times eight equals sixty-four.
If I were to tell you twenty-five squared,
that's twenty-five times twenty-five, which equals six hundred and twenty-five.
Powers of two is always very interesting.
It's especially interesting
if you one day go into computer science.
So two to the fourth power,
that's two times two times two times two.
So two times two is four so this equals sixteen.
And I did something very interesting here
kind of on purpose.
Notice that two to the fourth is equal to four times four, right?
Because we did four times four here.
I'm going to detail this more later on,
but I want to think about what that means.
Because four itself is the same thing as two squared.
So we learned, just real fast,
two to the fourth is the same thing
as two squared times two squared.
So I'll let you sit and think about that,
but other than that I think you have the general idea
of how basic exponents work,
and I think you're ready to try the level one exponent module.
have fun.