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# 用到積的指數性質 (英): 用到積的指數性質

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- 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.

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