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# 用到商的指數性質 (英): 用到商的指數性質

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- Let's do some exponent examples
- that involve division.
- Let's say I were to ask you what 5 to the sixth power
- divided by 5 to the second power is?
- Well, we can just go to the basic definition of what an
- exponent represents and say 5 to the sixth power, that's
- going to be 5 times 5 times 5 times 5 times 5-- one
- more 5-- times 5.
- 5 times itself six times.
- And 5 squared, that's just 5 times itself two times, so
- it's just going to be 5 times 5.
- Well, we know how to simplify a fraction or a rational
- expression like this.
- We can divide the numerator and the denominator by one 5,
- and then these will cancel out, and then we can do it by
- another 5, or this 5 and this 5 will cancel out.
- And what are we going to be left with?
- 5 times 5 times 5 times 5 over 1, or you could say that this
- is just 5 to the fourth power.
- Now, notice what happens.
- Essentially we started with six in the numerator, six 5's
- multiplied by themselves in the numerator, and then we
- subtracted out.
- We were able to cancel out the 2 in the denominator.
- So this really was equal to 5 to the sixth power minus 2.
- So we were able to subtract the exponent in the
- denominator from the exponent in the numerator.
- Let's remember how this relates to multiplication.
- If I had 5 to the-- let me do this in a different color.
- 5 to the sixth times 5 to the second power, we saw in the
- last video that this is equal to 5 to the 6 plus-- I'm
- trying to make it color coded for you-- 6 plus 2 power.
- Now, we see a new property.
- And in the next video, we're going see that these aren't
- really different properties.
- They're really kind of same sides of the same coin when we
- learn about negative exponents.
- But now in this video, we just saw that 5 to the sixth power
- divided by 5 to the second power-- let me do it in a
- different color-- is going to be equal to 5 to the-- it's
- time consuming to make it color coded for you-- 6 minus
- 2 power or 5 to the fourth power.
- Here it's going to be 5 to the eighth.
- So when you multiply exponents with the same base, you add
- the exponents.
- When you divide with the same base, you subtract the
- denominator exponent from the numerator exponent.
- Let's do a bunch more of these examples right here.
- What is 6 to the seventh power divided by 6
- to the third power?
- Well, once again, we can just use this property.
- This going to be 6 to the 7 minus 3 power, which is equal
- to 6 to the fourth power.
- And you can multiply it out this way like we did in the
- first problem and verify that it indeed will be 6 to the
- fourth power.
- Now let's try something interesting.
- This will be a good segue into the next video.
- Let's say we have 3 to the fourth power divided by 3 to
- the tenth power.
- Well, if we just go from basic principles, this would be 3
- times 3 times 3 times 3, all of that over 3 times 3-- we're
- going to have ten of these-- 3 times 3 times 3 times 3
- times 3 times 3.
- How many is that?
- One, two, three, four, five, six, seven, eight, nine, ten.
- Well, if we do what we did in the last video, this 3 cancels
- with that 3.
- Those 3's cancel.
- Those 3's cancel.
- Those 3's cancel.
- And we're left with 1 over-- one, two, three,
- four, five, six 3's.
- So 1 over 3 to the sixth power, right?
- We have 1 over all of these 3's down here.
- But that property that I just told you, would have told you
- that this should also be equal to 3 to the 4 minus 10 power.
- Well.
- What's 4 minus 10?
- Well, you're going to get a negative number.
- This is 3 to the negative sixth power.
- So using the property we just saw, you'd get 3 to the
- negative sixth power.
- Just multiplying them out, you get 1 over 3
- to the sixth power.
- And the fun part about all of this is these
- are the same quantity.
- So now you're learning a little bit about what it means
- to take a negative exponent.
- 3 to the negative sixth power is equal to 1 over 3 to the
- sixth power.
- And I'm going do many, many more examples of this in the
- next video.
- But if you take anything to the negative power, so a to
- the negative b power is equal to 1 over a to the b.
- That's one thing that we just established just now.
- And earlier in this video, we saw that if I have a to the b
- over a to the c, that this is equal to a to the b minus c.
- That's the other property we've been using.
- Now, using what we've just learned and what we learned in
- the last video, let's do some more complicated problems.
- Let's say I have a to the third, b to the fourth power
- over a squared b, and all of that to the third power.
- Well, we can use the property we just learned to simplify
- the inside.
- This is going to be equal to-- a to the third
- divided by a squared.
- That's a to the 3 minus 2 power, right?
- So this would simplify to just an a.
- And you could imagine, this is a times a times a
- divided by a times a.
- You'll just have an a on top.
- And then the b, b to the fourth divided by b, well,
- that's just going to be b to the third, right?
- This is b to the first power.
- 4 minus 1 is 3, and then all of that in parentheses to the
- third power.
- We don't want to forget about this third power out here.
- This third power is this one.
- Let me color code it.
- That third power is that one right there, and then this a
- in orange is that a right there.
- I think we understand what maps to what.
- And now we can use the property that when we multiply
- something and take it to the third power, this is equal to
- a to the third power times b to the third
- to the third power.
- And then this is going to be equal to a to the third power.
- times b to the 3 times 3 power, times b to the ninth.
- And we would have simplified this about as
- far as you can go.
- Let's do one more of these.
- I think they're good practice and super-valuable
- experience later on.
- Let's say I have 25xy to the sixth over 20y
- to the fifth x squared.
- So once again, we can rearrange the numerators and
- the denominators.
- So this you could rewrite as 25 over 20 times x over x
- squared, right?
- We could have made this bottom 20x squared y to the fifth--
- it doesn't matter the order we do it in-- times y to the
- sixth over y to the fifth.
- And let's use our newly learned exponent properties in
- actually just simplify fractions.
- 25 over 20, if you divide them both by 5, this is
- equal to 5 over 4.
- x divided by x squared-- well, there's two ways you could
- think about it.
- That you could view as x to the negative 1.
- You have a first power here.
- 1 minus 2 is negative 1.
- So this right here is equal to x to the negative 1 power.
- Or it could also be equal to 1 over x.
- These are equivalent.
- So let's say that this is equal into 1 over
- x, just like that.
- And it would be. x over x times x.
- One of those sets of x's would cancel out and you're just
- left with 1 over x.
- And then finally, y to the sixth over y to the fifth,
- that's y to the 6 minus 5 power, which is just y to the
- first power, or just y, so times y.
- So if you want to write it all out as just one combined
- rational expression, you have 5 times 1 times y, which would
- be 5y, all of that over 4 times x, right?
- This is y over 1, so 4 times x times 1, all of that over 4x,
- and we have successfully simplified it.

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