指數法則 1 (英)

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指數法則 1 (英) : 介紹指數法則

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Welcome to the presentation on level one exponent rules.
Let's get started with some problems.
So if I were to ask you what two--
that's a little fatter than I wanted it to be,
but let's keep it fat so it doesn't look strange--
two to the third times --
and dot is another way of saying times--
if I were to ask you what two to the third times two to the fifth is,
how would you figure that out?
Actually, let me use a skinnier pen because that does look bad.
So, two to the third times two to the fifth.
Well there's one way that I think you do know how to do it.
You could figure out that two to the third is eight,
and that two to the fifth is thirty-two.
And then you could multiply them.
And eight times thirty-two is two hundred forty plus sixteen, it's two hundred fifty-six, right?
You could do it that way.
And that's reasonable,
because it's not that hard to figure out what two to the third is and what two to the fifth is.
But if those were much larger numbers this method might become a little difficult.
So I'm going to show you, using exponent rules, you can actually multiply exponentials or exponent numbers
without actually having to do as much arithmetic.
Or actually you could handle numbers much larger than your normal math skills might allow you to.
So let's just think what two to the third times two to the fifth means.
Two to the third is two times two times two, right?
And we're multiplying that times two to the fifth.
And that's two times two times two times two times two.
So what do we have here?
We have two times two times two,
times,
two times two times two times two times two.
Really all we're doing is we're multiplying two how many times?
Well, one, two, three, four, five, six, seven, eight.
So that's the same thing as two to the eighth.
Interesting.
Three plus five is equal to eight.
And that makes sense because two to the three is two multiplying by itself three times,
to the fifth is two multiplying by itself five times,
and then we're multiplying the two,
so we're going to multiply two eight times.
I hope I achieved my goal of confusing you just now.
Let's do another one.
If I said seven squared times seven to the fourth.
That's a four.
Well, that equals seven times seven, right, that's seven squared,
times and now let's do seven to the fourth.
Seven times seven times seven times seven.
Well now we're multiplying seven by itself six times,
so that equals seven to the sixth.
So in general, whenever I'm multiplying exponents of the same base, that's key,
I can just add the exponents.
So seven to the one hundredth power times seven to the fiftieth power--
and notice this is an example now--
It would be very hard without a computer to figure out what seven to the one hundredth power is.
And likewise, very hard without a computer to figure out what seven to the fiftieth power is.
But we could say that this is equal to seven to the one hundred plus fifty,
which is equal to seven to the one hundred fifty.
Now I just want to give you a little bit of warning,
make sure that you're multiplying.
Because if I had seven to the one hundred plus seven to the fifty,
there's actually very little I could do here.
I couldn't simplify this number.
But I'll throw out one to you.
If I had two to the eighth times two to the twentieth.
Well, we know we can add these exponents.
So that gives you two to the twenty-eighth, right?
What if I had two to the eighth plus two to the eighth?
This is a bit of a trick question.
Well I just said if we're adding, we can't really do anything.
We can't really simplify it.
But there's a little trick here that we actually have two two to the eighths, right?
There's two to the eighth times one, two to the eighth times two.
So this is the same thing as two times two to the eighth, isn't it?
Two times two to the eighth.
That's just two to the eighth plus itself.
And two times two to the eighth,
well that's the same thing as two to the first times two to the eighth.
And two to the first times two to the eighth, by the same rule we just did, is equal to two to the ninth.
So I thought I would just throw that out to you.
And it works even with negative exponents.
If I were to say five to the negative one hundred times three to the say, one hundred
oh sorry, times five -- this has to be a five, too.
I don't know what my brain was doing.
Five to the negative one hundred times five to the one hundred two,
that would equal five squared, right?
I just take minus one hundred plus one hundred two.
This is a five.
I'm sorry for that brain malfunction.
And of course, that equals twenty-five.
So that's the first exponent rule.
Now I'm going to show you another one,
and it kind of leads from the same thing.
If I were to ask you what two to the ninth over two to the tenth equals--
Wow! That looks like that could be a little confusing.
But it actually turns out to be the same rule.
Because what's another way of writing this?
Well, we know that this is also the same thing as two to the ninth
times one over two to the tenth, right?
And we know one over two to the tenth.
Well, you could rewrite right this as two the ninth
times two to the negative ten, right?
All I did is I took one over two to the ten and I flipped it
and I made the exponent negative.
And I think you know that already from level two exponents.
And now, once again, we can just add the exponents.
Nine plus negative ten equals two to the negative one,
or we could say that equals one half, right?
So it's an interesting thing here.
Whatever is the bottom exponent, you could put it in the numerator like we did here,
but turn it into a negative,
So that leads us to the second exponent rule,
a simplification is we could just say that this equals two to the nine minus ten,
which equals two to the negative one.
Let's do another problem like that.
If I said ten to the two hundredth over ten to the fiftieth,
well that just equals ten to the two hundred minus fifty, which is one hundred fifty.
Likewise, if I had seven to the fortieth power over seven to the negative fifth power,
this will equal seven to the fortieth minus negative five.
So it equals seven to the forty-fifth.
Now I want you to think about that, does that make sense?
Well, we could have rewritten this equation as
seven to the fortieth times seven to the fifth, right?
We could have taken this one over seven to the negative five and turned it into seven to the fifth,
and that would also just be seven to the forty-five.
So the second exponent rule I just taught you actually is no different than that first one.
If the exponent is in the denominator,
and of course, it has to be the same base and you're dividing,
you subtract it from the exponent in the numerator.
If they're both in the numerator,
as in this case: seven to the fortieth times seven to the fifth --
actually there's no numerator, but if they're essentially multiplying by each other,
and of course, you have to have the same base--
then you add the exponents.
I'm going to add one variation of this, and actually this is the same thing,
but it's a little bit of a trick question.
What is two to the ninth times four to the one hundredth?
Actually, maybe I shouldn't teach this to you.
You'll have to wait until I teach you the next rule.
But I'll give you a little hint.
This is the same thing as two the ninth times two squared to the one hundredth.
And the rule I'm going to teach you now is that when you have something to an exponent,
and then that number raised to an exponent,
you actually multiply these two exponents.
So this would be two the ninth times two to the two hundredth.
And by that first rule we learned,
this would be two to the two hundred ninth.
Now in the next module I'm going to cover this in more detail.
I think I might have just confused you.
But watch the next video
and then after the next video I think you're going to be ready to do level one exponent rules.
Have a lot of fun!
Welcome to the presentation on level one exponent rules. Let's get started with some problems.
So if I were to ask you what 2 that's a little fatter than I wanted it to be but let's just keep it fat s
so it doesn't look strange.
two to the third times and dot is another way of saying times. If I were to ask you what two to the
third times
two to the fifth is
how would you figure that out?
actually let me use a skinnier pen
because that doesn't look
so two to the third times two to the fifth
well, there's one way that I think you do know how to do it
you could figure out that two to the third is eight
and that two to the fifth is thirty-two
and then you could multiply them
and what