![]() ![]() In many ways the exponent rule fits with the other exponent patterns very very well. ![]() And you see this same pattern continues perfectly for both the positive and negative numbers. Take another step, 2 to the -3 and one-eighth. Now take another step, that would be 2 to the -2 equals one-half divided by 2 which would be one-quarter. So in the top row that exponent would go down from zero to -1 and we would divide 1 by 2, so we would get one-half, 2 to the -1 equals one-half. Well again, going to the left the exponents go down by one each step in the top row and the numbers get divided by two each step in the bottom row. What would happen if we walk to the left of zero? So, here's our sidewalk again, but we've just extended it in, extended to the left past 2 to the 0. So, if we start at 2 to the 5th and 32, as we start taking steps to the left we're subtracting one from the exponent and we're dividing the purple number in the bottom by 2. Each step to the left we subtract one from the exponent and we divide by 2 in the bottom row. That's what happens when we move to the right. So to go from 1 to 2 to 4 to 8 to 16, each step we're multiplying by 2. So the exponent is increasing in the green row in the top and we multiply by a factor of two in the bottom row. So that's true for every box here, and as we move to the right, what's happening is we add one to the exponent. So notice that every number on the top equals the power on the top, equals the output on the bottom. Thinking about negative exponents, it's really good to have a variety of ways to make sense of it. It's good to have as many ways to think about this as possible, because it's a somewhat anti-intuitive idea. So this is another way to think about why b to -n = 1 / b to the n. Well, if we have subtraction in the exponents, that means divide the powers, that must mean b to the 0 divided by b to the n, and of course, b to the 0 = 1. So this means that b to the -n, we can think of that as b to the 0- n. In general, -n, we can write that as 0- n. Any negative number can be written as zero minus the absolute value of that number. Here's another way to think think about it. So that is the exponent rule, that is the rule for negative exponents and this is one way to think about it. If these two equal the same thing, they must equal each other, and this suggests that b to the -n equals 1 / b to the n. One way of thinking about it, we've got 13 to the -3 another way of thinking about it we've got 1 over 13 cubed. They're gonna cancel when we cancel we're gonna be left with one in the numerator and we're gonna have three factors of 13 in the denominator and of course that would be 1/13 cubed. And so what we have here of course we're gonna get some cancellation, we're gonna cancel four of those factors of 13 in the numerator and denominator. And similarly in the denominator, we have seven factors of 113 multiplied together. The fundamental definition of an exponent is that 13 to the 4 means that we're multiplying four factors of 13 together. Now let's go back and think about this in terms of the fundamental definition of an exponent. Well, if we just follow the division of the powers pattern for exponents, of course that tells it to subtract the own exponents, we'd get, 13 to the 4 minus 7, or 13 to the -3, all right? That's one way to approach this. Well, the power in the denominator is clearly a larger power. So for example, suppose we had 13 to the 4 divided by 13 to the 7. Let's look at a numerical example with a higher power in the denominator. We know that if we made the denominator exponent bigger and the numerator exponent smaller, then we would get a negative result for the subtraction and that would give us a negative exponent. In this particular case, we know the division rule for powers, that's something we've talked about in the previous video. ![]() This is something that happens in mathematics over and over again. And we have to ask ourselves exactly what would this mean, what would it mean to have a negative integer in the exponent? We have b to the- 3, what on earth would that mean? Well, as mathematicians often do, we will take a pattern that we already know and understand and extend it to cover something not yet covered by the rules. Now we're moving a little bit outside of that, we're expanding the definition, where the exponent can be a negative integer as well. And so, we think of the exponent as something we can count. ![]() So really we've kinda stuck with this idea that an exponent means the number of factors multiplied together. So notice so far in these lessons, we have discussed only positive integer exponents and zero as an exponent. At this point we are ready to talk about the idea of negative exponents. ![]()
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