In this video, we're going to talk about some different patterns of exponential growth. 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. Read full transcript
What's important to get from this video are the patterns. What's getting bigger, what's 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 in 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 1, we're gonna have a positive base greater than one. We have already seen this in the last video. I will use powers of 7 as an example. 7 to the 1 is 7.
7 squared is 49. I mentioned in the last video that 7 cubed is 343. That's a good number to know and then as we get to higher powers of 7, 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 5 or greater than 10, then what's gonna happen as 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 1 is one-half.
One half squared is a quarter, then an eighth, then a sixteenth. Notice that things are getting smaller, and smaller, and smaller. We get down to 1 over 128, and finally down to 1 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're gonna 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 3. 3 to the 1 is 3. Negative 3 squared is positive 9.
Negative 3 cubed is negative 27. Negative 3 to the 4th is positive 81. Negative 3 to the 5th, we talked about this a little on the last video. 3 to the 5th is 243. So negative 3 to the 5th is negative 243, and negative 3 to the 6 is positive 729. So again, notice we have this alternating pattern.
We saw this alternating pattern in the previous video. The absolute values of these powers are getting bigger each time, but the positive/negative sides are alternating. So this combines the idea of case one with continuously getting bigger. What's continuously getting bigger are the absolute 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 is happening when we raised 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 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 take negative one-half.
Negative one-half to the 1 is of course, negative one-half. Negative one squared is positive one-quarter. Negative one cubed is negative one-eighth. Negative 1 to the 4th is positive 16. Negative one-half to the 5th is negative 32. Negative 1 to the 6th is positive 64.
Negative one-half to the 7th is negative 128. And then, positive 1 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 to the 7th greater than x to the 6th? Well there's 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, and that's also a no answer. x to the seventh would not be greater than x to the sixth if it's equal to x to the sixth.
Now consider this question, if x is less than one and x is unequal to zero, is x to the seventh greater than x to the sixth? Well, we have to consider what happens in different cases. First of all, it is 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, no. x to the sixth 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 two-thirds, square that we get four-ninths. If we cube it, we get eight-27ths. Now notice that four-ninth, two-third is above one-half, four-ninths is slightly below one-half, it's above one-quarter. Eight-27ths.
Well eight-27ths is definitely less than a third. Four-ninths is greater than a third. Eight-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 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 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 if we extend this pattern, of course x to the 6th is gonna be bigger than x to the 7th. This is also gonna produce a no answer, and it turns out 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.