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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

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