The world of exponents
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Level 1 Exponents
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Understanding Exponents 2
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Understanding Exponents
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Positive and zero exponents
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Level 2 Exponents
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Negative Exponent Intuition
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Negative exponents
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Zero, Negative, and Fractional Exponents
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Level 3 exponents
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Evaluating exponential expressions
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Fractional exponents
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Negative fractional exponents
Level 2 Exponents negative exponents
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- Welcome to the presentation on the Level two exponents.
- In Level two exponents,
- the only thing we're going to add to the mix now
- is the concept of a negative exponent.
- So we learned already that 2 to the third power,
- well, that just equals 2 x 2 x 2.
- Hopefully, by now, that's second nature to you,
- and that equals 8.
- Now I'm going to teach you
- what 2 to the negative third power is.
- I know a lot of you are going to say, oh no, that's -8,
- but whenever you see exponents,
- you always have to remind yourself,
- exponents are not multiplication.
- I know there's that temptation to say,
- well, 2 x -3 is -6,
- so maybe 2 to the negative third power is -8,
- but that's not the case.
- And I'll explain in a future module
- why we use this convention
- but 2 to the negative 3rd power, it turns out,
- is equivalent to 1/2 to the 3rd power.
- So it turns out that the negative exponent,
- what it does is it means to take the inverse of the base,
- we'll call the number 2 as the base--
- and take that to the positive version of the exponent.
- And 1/2 to the third power?
- Well, we already learned that's 1/2 x 1/2 x 1/2,
- and that equals 1/8.
- So we say 2 to the negative third power.
- That didn't come out right.
- 2 to the negative third power is equal to 1/8.
- Let's do another one.
- Let's say 3 to the -2 power.
- Once again, immediately when we see
- that negative in the exponent, the easiest thing to do
- is just immediately take the reciprocal of the base.
- So we take 1/3.
- And we raise that to the +2 power.
- And that's easy now.
- 1/3 squared, well, that's equal to 1/9.
- Let's do some more problems.
- What if I had 2/3 to the negative third power?
- Once again, just to make it simple, whenever I see
- that negative in the exponent, I want to get rid of it.
- So I immediately take the reciprocal of the base.
- The reciprocal of 2/3 is 3/2 ,
- and I raise that to the positive third power.
- So what changed between the left and the right side?
- The 2/3 I flipped,
- and I turned the -3 into a +3.
- And now this just becomes a Level one exponent.
- This equals 3/2 x 3/2 x 3/2, and that equals 27/8.
- So that's interesting.
- 2/3 to the -3 is equal to 27/8.
- Let's do some more.
- Let's do 4/7 to the -1.
- Once again, we have a negative number in the exponent.
- That's the same thing as taking the reciprocal of the base
- and raising it to the positive exponent.
- Well, 7/4 to the 1, any number to the first power
- is just the same number.
- It's equal to 7/4.
- So when take it to the -1 power, all you're essentially doing
- is getting the reciprocal of the number.
- Let's do some more problems.
- 2 to the -5.
- Once again, we take the reciprocal of 2,
- and we say 1/2
- and now that can be raised to the 5th power,
- and that equals 1/2 x 1/2 x 1/2 x 1/2 x 1/2,
- and that equals 1/32.
- Another way we could have viewed 2 to the negative 5th
- is that, 2 to the negative 5th,
- we could have said that equals 1/2 to the 5th.
- We know that 2 to the 5th is 32.
- That would be the same thing.
- Two ways to do it,
- pretty much just changing the order of when you flip
- versus when you actually calculate the exponent.
- Let me do two or three more problems.
- And after I write down each of these problems,
- you might just want to pause it
- and see if you can do the problem yourself,
- and then compare your answer to mine.
- So let's say I had -4 to the negative 3rd power.
- Immediately, I like to get rid of the negative in the exponent,
- and I know that that equals -1/4 to the 3rd power,
- and that equals -1/4 x -1/4
- x -1/4.
- The negative times a negative is a positive,
- but then we're multiplying that times another negative,
- so we get a negative.
- 1 x 1 x 1 is 1.
- 4 x 4 is 16 x 4 is 64.
- So it equals -1/64.
- Let's do another problem.
- Let me think of a good number.
- 8/9.
- Let's make it negative.
- -8/9 to the negative 2nd power.
- Well, once again, that equals -9/8.
- Notice, I immediately just took the reciprocal of the base
- to the +2 power,
- and now that equals -9/8 x -9/8.
- A negative times a negative is a positive,
- so we get 9 x 9 is 81/64.
- I think you get the point now.
- The only new thing we've learned, really, is that
- when you have a negative exponent,
- it's the same thing as taking the reciprocal of the base
- and raising it to the positive exponent.
- Hopefully, that last statement didn't confuse you more,
- did more good than damage,
- but I think you're ready to try some problems now.
- Have fun.
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