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Thinking more about negative exponents

Video transcript
In the last video, we gave an argument for why 3 to the negative 2 power should be equivalent to 1 over 3 to the positive 2 power. I now want to do a very similar argument, but I want to do it for general bases and general exponents. So in particular, I want to think about what a to the b power, or in particular, what a to the negative b power should be equal to. So we know that we want to uphold the property that, if we have the base raised to an exponent and then that's being multiplied by that same base raised to another exponent, that we would just add the two exponents. So just to be clear, we want to maintain this property, that a to the b power times a to the negative b power should be equal to a to the b plus negative b power, which of course is going to be equal to a to the 0-th power. And as long as a does not equal to 0, this should be equal to 1. Let me write that-- for a does not equal 0. So in general-- actually I'll just copy and paste this-- a to the b times a to the negative b-- copy and paste-- should be equal to 1. So this right over here needs to be equal to 1. And so if we want think about, what does a to the negative b equal? We just divide both sides by a to the b. And we would get a to the negative b. I've kept the colors consistent this long. Gotta keep going. a to the negative b is equal to 1 over a to the b. And so I just want to review this. And it's really important to keep in mind because when you see something as a base to a negative exponent, there's a temptation to somehow introduce a negative number. But a negative exponent is really about taking the reciprocal. So you're taking the reciprocal of this number raised to the positive base. And we've just shown the argument that if we want this property be true-- and a lot of the properties of exponents are all about just keeping consistency, keeping the old properties continuing to be true as we introduce new definitions and new properties-- and we see that if we want this to be true, then a to the negative b should be 1 over a to the b. And in other videos we'll show other motivations for why a negative exponent is essentially defined this way. But now that we've seen this is true, for any non-zero a and any b, let's actually do some examples. And actually, let's think about the size of these numbers. So for example-- and let me introduce new colors right over here. I said I would introduce new colors, and I didn't-- if I were to take 5 to the third power, we know that's 5 times 5 times 5. That is equal to 125. It's a reasonably large number. Now what should 5 to the negative 3 power be? And I encourage you to pause this video and think about this before I tell you. Use what I just told you. What should 5 to the negative 3 power be? Well, I assume you've paused the video. So this is going to be equal to 1 over 5 to the positive 3 power, which is equal to 1/125. So this is really interesting. 5 to the third was a reasonably large number. While 5 to the negative 3, it wasn't a negative number. It's still a positive number, but it's a very small number. It's 1/125. Let's do two more examples. Let's say I were to take negative 1/2, and I were to raise this to the third power. What is that going to be? Well, it's negative 1/2 times negative 1/2 times negative 1/2, which is, of course, going to be equal to negative 1/8. Now what should negative 1/2 to the negative 3 power be, based on everything that we just talked about? And once again, I encourage you to pause this video and think about it. Well it should be equal to 1 over negative 1/2 to the positive 3 power. Well, we just figured out what that is. Negative 1/2 to the positive 3 power is negative 1/8. This thing right over here should be equal to negative 1/8. So it's going to be equal to 1 over negative 1/8. And what's that equal to? Well that's equal to 1 times the reciprocal of this denominator over here. So it's 1 times negative 8/1, which is going to be equal to negative 8. So this is interesting. Normally when you take a number whose absolute value is less than 1 and you take it to larger and larger exponents, it gets smaller and smaller and smaller. When you multiply either a positive or negative 1/2 times 1/2 times 1/2, its magnitude, its absolute value, is normally going to get smaller and smaller and smaller. But now, when you raise it to a negative exponent, it's absolute value is getting larger and larger and larger. And that's because, once again, you are taking the reciprocal. The important thing I want to emphasize here is that you shouldn't view this negative as when you think about negative multiplication. Notice, this negative did not change the sign of what the eventual value is. You've got negative 1/8. You got negative 8. What it did is, it changed the magnitude. So this is negative 1/8. When you took to the negative third power, you took the reciprocal of that, which is negative 8.