Level 3 exponents fractional exponents
Level 3 exponents
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- Welcome to level three exponents.
- Let's get started.
- So if I asked you what four to the one / two power is, your immediate
- inclination is to view this probably as like a
- multiplication problem and try to multiply it somehow or add
- them or something.
- And you always have to remind yourself this
- is not multiplication.
- I know when I first learned it, I always was tempted to do
- something with multiplication.
- Well, something to the one / two power might not be intuitive to
- you, but it actually turns out that this means the same thing
- as the square root of four.
- Or another way, what times itself is equal to four,
- and we know that the square root of four is two.
- It could actually be a positive two or a negative two because we
- know that either of those numbers when they're
- squared could equal four.
- But for the sake of this one we'll assume it's always
- the positive square root.
- So four to the ½ is equal to two.
- Similarly, nine to the one / two, well that would be three.
- sixteen to the one / two -- oops.
- sixteen to the one / two, my subconscious gave away the answer.
- sixteen to the one / two power is four.
- twenty-five to the one / two power is five.
- I think that might make sense to you now.
- So what does it mean when something is to the one / three power?
- Well, if I say eight to the one / three power, you might
- already catch on.
- In the one / two power we said something times something
- is equal to four.
- Well in the one / three power we have to say that something to the
- third power is equal to eight.
- And if you've been practicing your exponents you know that two
- to the third power is equal to eight.
- So we know that eight to the one / three is equal to two.
- Similarly, twenty-seven to the one / three is equal to three.
- And sixty-four to the one / three is equal to four.
- You might notice that I'm picking particular numbers,
- like eight and twenty-seven, sixty-four.
- That's because they have clean, cube roots.
- And then by the way, when something is to the one / three power
- that's the same thing as saying the cube root.
- I just used terminology without explaining it,
- which is very bad.
- So I just used eight and twenty-seven and sixty-four because when I raised them to
- the fractional exponents, they actually come out to
- be clean numbers.
- You could use a calculator and do something like five to the one / three
- power and you'll get some weird decimal.
- Let's do some more problems.
- So we know that nine to the one / two is equal to three.
- Well what do you think nine to the two / three is equal to?
- Well, it turns out that this is equivalent to nine to the -- oops,
- I actually didn't want to do those -- what do you think nine
- to the three / two is equivalent to?
- Well this is the same thing as nine to the one / two power
- to the third power.
- And I'll do a whole presentation on the actual
- principles of exponents, but it actually turns out you
- could just multiply.
- When you have one exponent to another is when you can
- multiply the two and that's where you get three / two .
- But nine to the one / two we know is three.
- And you're raising it to the third power, so that equals twenty-seven.
- I'm sure at this point I have confused you.
- Let's do more of these.
- So you know at this point that sixteen to the one / four power --
- think about what that is.
- That means that some number to the fourth power is sixteen.
- If you've been practicing your level one exponents, you'll
- probably know that well, that equals two, because two times two
- times two times two, well that equals sixteen.
- So we know that sixteen to the one / four is equal to two.
- So what do you think sixteen to the two / four is equal to?
- Well, we already know from that last problem that that's the
- same thing as sixteen to the one / four squared -- that's the two on both
- sides -- and we know sixteen to the one / four is two, so that equals two
- squared and that equals four.
- And it all works out because we know from fractions another way
- to write the fraction two / four is to write one / two.
- So this is the same thing as sixteen to the one / two power.
- sixteen to the one / two power, well that's just equal to four.
- Now I'm going to mix it up real good and do some negative
- fractional exponents.
- So what if I were to tell you sixteen to the negative one / two power?
- Well this might seem very daunting at first, but as we
- know with the negative exponents level three,
- immediately we just say well this is the same thing as one
- over sixteen to the positive one / two.
- And that's the same thing as one to the one / two over sixteen to the one / two.
- Well the square root of one is easy, it's one.
- And sixteen to the one / two is four.
- So that wasn't too bad.
- It's a little daunting when you see a negative exponent, but
- immediately when you see that negative, just flip the sixteen and
- then work it out like a regular fractional exponent problem.
- Let's do another one.
- eight over twenty-seven to the negative one / three.
- Immediately when we see that negative, we
- want to just flip it.
- So we'll say that equals twenty-seven over eight to the one / three, and that
- equals twenty-seven to the one / three over eight to the one / three.
- And we know that twenty-seven to the one / three, well that equals three.
- And eight to the one / three, well that equals two.
- So we've got eight over twenty-seven to the negative one / three is three / two.
- The first problem probably looked very intimidating to
- you, but it only took us two steps to get there and as you
- do more practice, hopefully it'll seem more and
- more intuitive to you.
- Let me give you another problem.
- What's negative eight to the negative third power?
- Let me change that.
- What's negative eight to the negative one / three power?
- Once again, at first this might confuse you, but when you see
- that negative in the exponent, we just take the reciprocal of
- the base, so that we say that that is equal to negative one
- over eight to the one / three power.
- And we say well that is equal to, we could write it as one
- over negative eight to the one / three.
- We say what number times itself three times is equal to negative eight?
- Well, we know from intuition there's no real mechanical way
- to do this, but we know that negative two times negative two is
- four times negative two is negative eight.
- So we know that this is equal to one over negative
- two or negative one / two.
- So negative eight to the negative one / three is
- equal to minus one / two.
- Let's do another one, one more problem just to thoroughly
- melt your brain.
- Let's say nine over four to the negative three over two.
- Well, immediately we see that negative exponent,
- let's flip the base.
- We get four over nine to the three / two .
- Well we know that that equals four over nine to the one / two, and all
- of that to the third power.
- four over nine to the one / two, I think at this point you know that's the
- same thing as four to the one / two which is two over nine to
- the one / two which is three.
- Now we have to just raise everything to the third power.
- And that's the same thing as two the third power which is
- eight divided by three to the third power.
- Well that's twenty-seven.
- There we have nine / four to the negative three / two power is
- equal to eight over twenty-seven.
- Now hopefully you can at least do these problems.
- You probably don't have a good intuitive sense for exactly
- what a negative three / two power is, and hopefully I can cover that
- for you in future modules.
- But I think you're ready to try some of the level three
- exponent practice problems.
- Have fun.
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