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Exponential Growth
In this video, we are 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 will emphasize in the 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. It's 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'll use powers of seven as an example seven to the one is seven. Seven squared is 49. I mentioned in the last video that cubed is 343.
That's a good number to know. And then as we get to higher powers of 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 base greater than one, and especially If it's greater than five or greater than ten then what's going to happen is you start raising powers of it, it's gonna get inconceivably big very quickly.
So the big idea is a positive base greater than one, the power 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 1/2 for example. So 1/2 to the 1 is 1/2. 1/2 squared is 1/4, then 1/8, then 1/16. Notice that things are getting smaller and smaller and smaller. We get down to one over 128, and finally 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 is possible for higher exponents to produce small patterns. 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're gonna be following a very different pattern for exponential growth, than if the base were more than one. Now, even more interested. Let's talk about a negative base less than -1.
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. (-3) squared = +9. (-3) cubed = -27. (-3) to the 4th is +81.
Negative three to the fifth, we talked about this little in the last video. Three to the fifth is 243 so negative three to the fifth is -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, negative signs are alternating.
So this combines the idea of case one with continuously getting bigger. What's continuously getting bigger are the absent values of the numbers. But the actual number itself is flip flopping between positive and 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 negative one and zero.
So this would be a number that is negative and has an absolute value less than one, it is between -1 and zero. So let's take negative one-half. Negative one-half to the 1 is, of course, negative one-half. Negative one squared is positive one-quarter. Negative 1 cubed is negative one-eighth.
Negative 1 to the fourth is positive 16. Negative one-half to the fifth is negative 32. Negative 1 to the sixth is positive 64. Negative one-half to the seventh is negative 128. And then positive one over 256. So notice Similar to what was happening 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 ask the question, is x^7 greater than x^6? Well, there's no clear answer. It would be true for positive numbers greater than one, and false for negatives. Also, if X = 0, x of the 7th would equal x of the 6th which would be zero. And that's also a no answer. x of the 7th would not be greater than x of the 6th if it's equal to x of the 6th.
Now, consider this question. If x<1, and x unequal 0, is x to the 7th > x to the 6th. Well 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 7th is negative and x to the 6th 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 6th is definitely gonna be bigger if x is negative. What if 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 2/3 is above 1/2. 4/9 is slightly below 1/2. It's above 1/4.
8 27ths, well 8 27ths is definitely less than a third. 4 9ths is greater than a third. 8 27ths is less than a third. Then we get to the 16 81st, that's actually less than a quarter. And what's happening is, is that these numbers are getting smaller and smaller. And this is what we've seen in these powers.
This is our case two above. Where we have positive numbers < 1. 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 6th is gonna be bigger than x to the 7.
This is also going to produce a no answer. And it turns out that 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 discussed the patterns of exponential growth, how increasing the exponent changes the size of the powers for different kinds of bases.
Read full transcriptWhat's important to get from this video are the patterns. What's getting bigger. What's getting smaller and when. It's 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'll use powers of seven as an example seven to the one is seven. Seven squared is 49. I mentioned in the last video that cubed is 343.
That's a good number to know. And then as we get to higher powers of 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 base greater than one, and especially If it's greater than five or greater than ten then what's going to happen is you start raising powers of it, it's gonna get inconceivably big very quickly.
So the big idea is a positive base greater than one, the power 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 1/2 for example. So 1/2 to the 1 is 1/2. 1/2 squared is 1/4, then 1/8, then 1/16. Notice that things are getting smaller and smaller and smaller. We get down to one over 128, and finally 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 is possible for higher exponents to produce small patterns. 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're gonna be following a very different pattern for exponential growth, than if the base were more than one. Now, even more interested. Let's talk about a negative base less than -1.
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. (-3) squared = +9. (-3) cubed = -27. (-3) to the 4th is +81.
Negative three to the fifth, we talked about this little in the last video. Three to the fifth is 243 so negative three to the fifth is -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, negative signs are alternating.
So this combines the idea of case one with continuously getting bigger. What's continuously getting bigger are the absent values of the numbers. But the actual number itself is flip flopping between positive and 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 negative one and zero.
So this would be a number that is negative and has an absolute value less than one, it is between -1 and zero. So let's take negative one-half. Negative one-half to the 1 is, of course, negative one-half. Negative one squared is positive one-quarter. Negative 1 cubed is negative one-eighth.
Negative 1 to the fourth is positive 16. Negative one-half to the fifth is negative 32. Negative 1 to the sixth is positive 64. Negative one-half to the seventh is negative 128. And then positive one over 256. So notice Similar to what was happening 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 ask the question, is x^7 greater than x^6? Well, there's no clear answer. It would be true for positive numbers greater than one, and false for negatives. Also, if X = 0, x of the 7th would equal x of the 6th which would be zero. And that's also a no answer. x of the 7th would not be greater than x of the 6th if it's equal to x of the 6th.
Now, consider this question. If x<1, and x unequal 0, is x to the 7th > x to the 6th. Well 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 7th is negative and x to the 6th 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 6th is definitely gonna be bigger if x is negative. What if 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 2/3 is above 1/2. 4/9 is slightly below 1/2. It's above 1/4.
8 27ths, well 8 27ths is definitely less than a third. 4 9ths is greater than a third. 8 27ths is less than a third. Then we get to the 16 81st, that's actually less than a quarter. And what's happening is, is that these numbers are getting smaller and smaller. And this is what we've seen in these powers.
This is our case two above. Where we have positive numbers < 1. 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 6th is gonna be bigger than x to the 7.
This is also going to produce a no answer. And it turns out that 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 discussed the patterns of exponential growth, how increasing the exponent changes the size of the powers for different kinds of bases.
