Negative Exponents
At this point we are ready to talk about the idea of negative exponents. So notice so far in these lessons, we have discussed only positive integer exponents and zero as an exponent. So really we've kinda stuck with this idea that an exponent means the number of factors multiplied together. And so, we think of the exponent as something we can count.
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 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.
This is something that happens in mathematics over and over again. In this particular case, we know the division rule for powers, that's something we've talked about in the previous video. 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.
Let's look at a numerical example with a higher power in the denominator. So for example, suppose we had 13 to the 4 divided by 13 to the 7. Well, the power in the denominator is clearly a larger power. 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. Now let's go back and think about this in terms of the fundamental definition of an exponent. The fundamental definition of an exponent is that 13 to the 4 means that we're multiplying four factors of 13 together. And similarly in the denominator, we have seven factors of 113 multiplied together.
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. 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. Now compare those two results.
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. 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. So that is the exponent rule, that is the rule for negative exponents and this is one way to think about it.
Here's another way to think think about it. Any negative number can be written as zero minus the absolute value of that number. For example, we could write -3 as 0- 3. In general, -n, we can write that as 0- n. So this means that b to the -n, we can think of that as b to the 0- 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.
And so this would be 1 / b to the n. So this is another way to think about why b to -n = 1 / b to the n. Here is another way to think about it. It's good to have as many ways to think about this as possible, because it's a somewhat anti-intuitive idea. Thinking about negative exponents, it's really good to have a variety of ways to make sense of it.
Imagine a sidewalk of exponents. So notice that every number on the top equals the power on the top, equals the output on the bottom. 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 the exponent is increasing in the green row in the top and we multiply by a factor of two in the bottom row.
So to go from 1 to 2 to 4 to 8 to 16, each step we're multiplying by 2. That's what happens when we move to the right. Each step to the left we subtract one from the exponent and we divide by 2 in the bottom row. 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.
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. 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. 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.
Now take another step, that would be 2 to the -2 equals one-half divided by 2 which would be one-quarter. Take another step, 2 to the -3 and one-eighth. Then 2 to the -4 and one-sixteenth. And you see this same pattern continues perfectly for both the positive and negative numbers.
In many ways the exponent rule fits with the other exponent patterns very very well. And the more you appreciate how they all fit together as a seamless whole the more you will really understand this rule. So the rule of course is b to the -n = 1 over b to the n. So another way to say this is, a base to a negative negative power is the reciprocal of that same base to the positive power.
This means that a negative exponent on a fraction will be the reciprocal to the positive power. So the fraction p over q to the -n, that will equal q / p to the positive n. That's a really handy shortcut to know on the test. A negative power in the numerator of a fraction can be moved to the denominator as a positive power, or likewise, from the denominator to the numerator.
So for example, if I have this fraction and I need to simplify it, well, that d to the -8 in the numerator, if I moved that to the denominator it will be a d to the positive 8. That h to the -4 in the denominator, if I move that to the numerator, it will become h to the positive 4. And so this is the expression now written with all positive powers.
So for example, you might be asked to simplify something like this. Pause the video and see if you can simplify this and then we'll talk about it. Well, of course the first thing we'll do is we can treat the numbers separately from the exponents and for the numbers we'll just factor out the greatest common factor which is 6.
So not changing the exponents at all, just the number is simplified to a 4/3. Now with that x to the -4 in the denominator, that can move up to the numerator. One way to think about it is it moves up to the numerator so we get an x to the 12th times an x to the 4th in the numerator. And other ways just to think about the law of divisions and we get an x to the 12 minus -4, and of course 12 minus -4 is the same as 12 plus 4, of course at the ys we just have ordinary division of powers.
So the ys, that's the easiest to handle. That is 9 minus 3, y to the 6 then we treat those xs. We move it up to the numerator, this is one way to handle it, we move it up to the numerator, of course then we add the powers and we get 4/3 x to the 16th, y to the 6th. And that is the most simplified we can make this.
Here's a practice problem, pause the video and then we'll talk about this. Okay, so we have to rank things from smallest to biggest, the test loves these ranking questions. So first of all, one-third to the -8. Well, what does that mean? That's the same as 3 to the positive 8.
Now remember that 3 to the 4th is 81. So let's approximate that as 80, 3 to the 4th is approximately 80. Well, 3 to the 8th is gonna be 3 to the 4th squared. So that's gonna be approximately 80 squared and 80 squared, that's up above 6,000. So that's a relatively large number, that's what one equals.
II, 3 to the -3. Well, this is one-third to the third, and so this is 1/27th, okay so that's clearly much smaller than 1. And then if we look at the last one, 1/3 to the 5th. Well, we don't even have to calculate the value. 1/3 to the 5th we know is going to be smaller than 1/3 to the 3.
And so that means that III is the smallest, II is the middle, and I is the biggest. So in order, it's III, II, I, and this is answer choice E. In summary, b to the -n = 1 / b to the n. A base to a negative exponent is one over the base to the positive of that exponent. A fraction to the -n equals to the reciprocal to the positive n so we can flip over a fraction and get rid of the negative in the exponent.
An exponent switch from negative to positive when we move them in a fraction from numerator to denominator or vice versa.
Read full transcriptNow 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 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.
This is something that happens in mathematics over and over again. In this particular case, we know the division rule for powers, that's something we've talked about in the previous video. 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.
Let's look at a numerical example with a higher power in the denominator. So for example, suppose we had 13 to the 4 divided by 13 to the 7. Well, the power in the denominator is clearly a larger power. 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. Now let's go back and think about this in terms of the fundamental definition of an exponent. The fundamental definition of an exponent is that 13 to the 4 means that we're multiplying four factors of 13 together. And similarly in the denominator, we have seven factors of 113 multiplied together.
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. 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. Now compare those two results.
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. 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. So that is the exponent rule, that is the rule for negative exponents and this is one way to think about it.
Here's another way to think think about it. Any negative number can be written as zero minus the absolute value of that number. For example, we could write -3 as 0- 3. In general, -n, we can write that as 0- n. So this means that b to the -n, we can think of that as b to the 0- 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.
And so this would be 1 / b to the n. So this is another way to think about why b to -n = 1 / b to the n. Here is another way to think about it. It's good to have as many ways to think about this as possible, because it's a somewhat anti-intuitive idea. Thinking about negative exponents, it's really good to have a variety of ways to make sense of it.
Imagine a sidewalk of exponents. So notice that every number on the top equals the power on the top, equals the output on the bottom. 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 the exponent is increasing in the green row in the top and we multiply by a factor of two in the bottom row.
So to go from 1 to 2 to 4 to 8 to 16, each step we're multiplying by 2. That's what happens when we move to the right. Each step to the left we subtract one from the exponent and we divide by 2 in the bottom row. 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.
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. 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. 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.
Now take another step, that would be 2 to the -2 equals one-half divided by 2 which would be one-quarter. Take another step, 2 to the -3 and one-eighth. Then 2 to the -4 and one-sixteenth. And you see this same pattern continues perfectly for both the positive and negative numbers.
In many ways the exponent rule fits with the other exponent patterns very very well. And the more you appreciate how they all fit together as a seamless whole the more you will really understand this rule. So the rule of course is b to the -n = 1 over b to the n. So another way to say this is, a base to a negative negative power is the reciprocal of that same base to the positive power.
This means that a negative exponent on a fraction will be the reciprocal to the positive power. So the fraction p over q to the -n, that will equal q / p to the positive n. That's a really handy shortcut to know on the test. A negative power in the numerator of a fraction can be moved to the denominator as a positive power, or likewise, from the denominator to the numerator.
So for example, if I have this fraction and I need to simplify it, well, that d to the -8 in the numerator, if I moved that to the denominator it will be a d to the positive 8. That h to the -4 in the denominator, if I move that to the numerator, it will become h to the positive 4. And so this is the expression now written with all positive powers.
So for example, you might be asked to simplify something like this. Pause the video and see if you can simplify this and then we'll talk about it. Well, of course the first thing we'll do is we can treat the numbers separately from the exponents and for the numbers we'll just factor out the greatest common factor which is 6.
So not changing the exponents at all, just the number is simplified to a 4/3. Now with that x to the -4 in the denominator, that can move up to the numerator. One way to think about it is it moves up to the numerator so we get an x to the 12th times an x to the 4th in the numerator. And other ways just to think about the law of divisions and we get an x to the 12 minus -4, and of course 12 minus -4 is the same as 12 plus 4, of course at the ys we just have ordinary division of powers.
So the ys, that's the easiest to handle. That is 9 minus 3, y to the 6 then we treat those xs. We move it up to the numerator, this is one way to handle it, we move it up to the numerator, of course then we add the powers and we get 4/3 x to the 16th, y to the 6th. And that is the most simplified we can make this.
Here's a practice problem, pause the video and then we'll talk about this. Okay, so we have to rank things from smallest to biggest, the test loves these ranking questions. So first of all, one-third to the -8. Well, what does that mean? That's the same as 3 to the positive 8.
Now remember that 3 to the 4th is 81. So let's approximate that as 80, 3 to the 4th is approximately 80. Well, 3 to the 8th is gonna be 3 to the 4th squared. So that's gonna be approximately 80 squared and 80 squared, that's up above 6,000. So that's a relatively large number, that's what one equals.
II, 3 to the -3. Well, this is one-third to the third, and so this is 1/27th, okay so that's clearly much smaller than 1. And then if we look at the last one, 1/3 to the 5th. Well, we don't even have to calculate the value. 1/3 to the 5th we know is going to be smaller than 1/3 to the 3.
And so that means that III is the smallest, II is the middle, and I is the biggest. So in order, it's III, II, I, and this is answer choice E. In summary, b to the -n = 1 / b to the n. A base to a negative exponent is one over the base to the positive of that exponent. A fraction to the -n equals to the reciprocal to the positive n so we can flip over a fraction and get rid of the negative in the exponent.
An exponent switch from negative to positive when we move them in a fraction from numerator to denominator or vice versa.
