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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 as an exponent? We had b to the negative three, 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 smaller, and the nu, 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 power pattern for exponents, of course that tells us to subtract the exponents.
We get 13 to the four minus seven or 13 to the negative three. 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 four means that we are multiplying four factors of 13 together.
And similarly in the denominator, we’d have seven factors of 13 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 on the numerator and the denominator, they’re gonna cancel. When we cancel we’re gonna be left with 1 in the numerator.
And we’re gonna have three factors of 13 in the denominator. And of course, it would be 1 over 13 cubed. And I’ll compare those two results. One way of thinking about it, we got 13 to the negative 3. Another way of thinking about it, we got 1 over 13 cubed. If these two equal the same thing, they must equal each other.
And this suggests that b to the negative n equals 1 over b to the n. So, that is the exponent rule. That is the rule for negative exponents. This is one way to think about it. Here’s another way to think about it. Any negative number can be written as zero minus the absolute value of that number.
For example, we could write negative 3 as zero minus positive 3. In general, negative n, we could write that as zero minus n. So this means that b to the negative n, we can think of that as b to the zero minus n. Well if we have subtraction in the exponents, that means divide the powers. That must mean b to the zero divided by b to the n and of course b to the zero is 1.
And so this would be 1 over b to the n. So this is another way to think about why b to the negative n equals one over b to the n. Here’s another way to think about it. It’s good to have as many ways to think about this as possible because it’s a, 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 1 to the exponent.
So the exponent is increasing in the green row at the top, and we multiply by a factor of 2 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 1 from the exponent, and we divide by 2 in the bottom row.
So as, if we start at 2 to the fifth and 32. As we start taking steps to the left we’re subtracting 1 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 just extended it, in, we extended it to the left past two to the zero.
Well again going to the left, the exponents go down by 1 each step in the top row, and the numbers get divided by 2 each step in the bottom row. So in the top row that exponent would go down from zero to negative 1 and we would divide 1 by 2 so we’d get one-half. 2 to the negative 1 equals one-half. Now take another step.
That would be two to the negative two equals one-half divided by two, which would be one quarter. Take another step. Two to the negative three and one-eighth. Then two to the negative four and one-16th. 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, to the negative n equals one over b to the n. So, another way to say this is a base to a 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 I have the fraction p over q to the negative n. That will equal q over 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 negative eight in the numerator. If I move that to the denominator it will be a d to the positive eight. That h to the negative four in the denominator if I move that to the numerator it will become h to the positive four. 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 six.
So, not changing the exponents at all. Just the numbers simplified to a 4 over 3. Now with that x to the negative 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.
Another way is just to think about the law of division. We get an x to the 12, minus negative 4. And of course 12 minus negative 4 is the same as 12 plus 4. Of course with the y, we just have ordinary division of powers. So the y, that’s the easiest to handle. That is nine minus three y to the six.
Then we treat those x. We move it up to the numerator. This is one way to handle it. We move it up to the numerator. Course then we add the powers and we get four thirds 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 negative 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 1 equals. Two, 3 to the negative 3rd. Well, this is one-third to the 3rd, and so this is 1-27th.
Okay, so, that’s clearly much smaller than 1. And then if we look at the last one, one third to the fifth, well we don’t even have to calculate the value. One third to the fifth we know is going to be smaller than one third to the, to the third. And so that means that three is the smallest, two is the middle and one is the biggest.
So in order it’s three, two, one. And this is answer choice E. In summary, b to the negative n equals 1 over b to the n. A base to a negative exponent is one over the base of, to the positive of that exponent. A fraction to the negative n equals the reciprocal, to the positive n.
So we could flip over a fraction and get rid of the negative in the exponent. And exponents switch from negative to positive when we move them in a fraction from numerator to denominator, or vice versa. .
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