رشد نماییفصل: بخش ریاضی / درس: ریشه ها و قدرت ها / درس 2
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In this video, we’re going to talk about some different patterns of exponential growth. The different patterns that we get when we have powers of different kinds of numbers. So, the first thing I’ll emphasize in this video, this video is not about the exact calculations. I would say worry less about the exact values of the number.
What’s important to get from this video are the patterns, what’s getting bigger, what’s getting smaller and when. And it’s very important to be aware of these properties in a variety of questions. The test absolutely loves these patterns and asks about them in several different ways.
For different bases, we will look at what happens to the powers when the exponents increase through the integers. So case one, we’re gonna have a positive base greater than one. We’ve already seen this in the last video. I will use powers of seven as an example. Seven to the one is seven.
Seven squared is 49. I mentioned in the last video that seven cubed is 343. That’s a good number to know. Then as we get to higher powers seven, these are not numbers that you need to know. I’m showing you these higher powers only to emphasize that exponential growth starts to get very big very quickly.
So this is one good idea to keep in mind. That if you have a, a base greater than one, and especially. If it’s, if it’s greater than five or greater than ten, then what’s gonna happen is you start raising powers of it. It’s gonna get inconceivably big very quickly. So the big idea here is a positive base greater than one, the powers continually get larger, at a faster and faster rate.
That’s very important. So that’s pattern number one. That’s the pattern when we have a positive base greater than one. Suppose we have a positive base less than one. Okay, well this is interesting. Let’s say one-half, for example.
So one-half to the, to the one is one-half. One-half squared is a quarter, then an eighth, then a 16th. Notice that things are getting smaller and smaller and smaller. We get down to one over 128 and finally down to one over 256. So we’ve gotten very, very small at this point. So much as in the first case, we got big very quickly, now we’re getting small very quickly.
It’s possible for higher exponents to produce smaller patterns. So in other words, as we raise the exponent higher and higher, it’s possible for the overall power to get smaller and smaller and smaller. And so this is an important thing to keep in mind. Numbers, when we have a base between zero and one, a positive base less than one. Then we are going to be following a very different pattern for exponential growth than if the base were more than one.
Now, even more interesting. Let’s talk about a negative base less than negative one. So this is a number that is negative and it has an absolute value more than one. So, for example, let’s just take three. Three to the one is three. Three to, negative three squared is positive nine.
Negative three cubed is negative 27. Negative three to the fourth is positive 81, negative three to the fifth, we talked about this a little in the last video, three to the fifth is 243, so negative three to the fifth is negative 243. And negative three to the sixth is positive 729. So again notice we have this alternating pattern.
We saw this alternating pattern in the previous video. The absolute values are getting, the absolute values of these powers are getting bigger each time. But the positive and negative signs are alternating. So this, this combines the idea of case one with continuously getting bigger. What’s continuous getting bigger are the absolute values of the numbers.
But the actual number itself, is flip flopping between positive, negative. So we get a big positive, then a bigger negative, then a bigger positive, then a bigger negative. It’s going back and forth like that. So, you can imagine these wild jumps on the number line. From a very large positive number to a very large negative number.
That’s what’s happening when we raise a negative base less than one to these powers. Finally, the last case, a negative fraction. That is to say, a negative base between zero and one, negative one and zero. So, this would be a number that is negative and has an absolute value less than one.
It is between negative one and zero. So, let’s taken negative one half. Negative one half to the one is of course negative one half. Negative one squared is positive one quarter. Negative one cubed is negative one-eighth, negative one to the 4th is positive 16. Negative one, negative one half to the 5th is negative 32, negative one to the 6th is positive 64, negative one half to the 7th is negative 128 and then positive one over 256.
So notice similar to what was happened in case two, the absolute values are getting smaller and smaller but we’re flip flopping again between positive and negative. So we’re getting closer to zero but we’re getting closer to zero by jumping back and forth above zero and below zero, we’re approaching zero. By this kind of skipping pattern going above it and below it and getting closer each time.
Notice that as the exponent increases, whether the power gets bigger or smaller depends on the base. So, we asked the question is x to the seventh greater than x to the sixth? Well there is no clear answer. It would be true for positive numbers greater than one and false for negatives. Also if x equals zero, x to the 7th would equal x to the 6th which would be zero.
Which is also a no answer, x to the 7th would not be greater than x to the 6th if it’s equal to x to the 6th. Now, consider this question. If x is less than one and x is unequal to zero is x to the 7th greater than x to the sixth? We have to consider what happens in different cases.
First of all it’s very easy to think about what happens with the negatives. If x is negative then x to the seventh is negative and x to the sixth is positive. And any positive is greater than any negative. So therefore we’re gonna get a no answer to the question. We’re gonna get a clear answer of no. X to the sixth is definitely gonna be bigger if x is negative.
What is x is a positive number between zero and one? So these are the only positive numbers allowed, the positive numbers between zero and one. Well if we square say 2/3, square that we get 4/9. If we cube it we get 8/27. Now notice that 4/9, 2/3 is, is above one half, 4/9 is slightly below one half.
It’s, it’s above one quarter 8/27, 8/27 is definitely less than 1/3. 4/9 is greater than 1/3. 8/27 is less than 1/3. Then we get to 16/81, that’s actually less than a quarter and what’s happening is that these numbers are getting smaller and smaller and this is what we’ve seen in these powers.
This is, this is what, this is our case two above where we have positive numbers less than one, as we raise higher and higher powers, we get smaller and smaller numbers. So we can definitely say that the powers are getting smaller, so we extend this pattern. Of course, x to the sixth is going to be bigger than x to the seventh.
This is also gonna produce a no answer. And it turns out the, the answer to the question is a consistent no for every x allowed. So we can give a definitive answer of no to this question. In this video we discuss the patterns of exponential growth. How increasing the exponent changes the size of the powers for different kinds of bases.
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