In continuing our series on Powers and Roots, we now ask, “what is exponential growth?” Watch the video below to continue learning these crucial math concepts. And don’t forget — the transcript to the video is below.
What Is Exponential Growth?
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 the 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.
Problem One: Pattern One
So case 1, we’re gonna have a positive base greater than one. We’ve already seen this in the last video, use powers of 7 as an example. 7 to the one 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 that is 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. We have a base greater than 1 and especially, if it’s greater than 5 or greater than 10, then what’s gonna 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 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?
Pattern Number Two
Well, this is interesting, let’s say, 1/2 for example. So 1/2 to the 1 = 1/2, 1/2 squared is a quarter, then an 8th, then a 16th. Notice that things are getting smaller and smaller and smaller. We get down to 1 / 128, and finally down to 1 / 256. So we’ve gotten very, very small at this point. 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 exponents 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 0 and 1, a positive base less than 1, then we’re gonna be following a very different pattern for exponential growth than if the base were more than 1.
Rules Involving a Negative Base
Now, even more interesting, let’s talk about a negative base less than -1. So this is a number that is negative and it has an absolute value of more than 1. For example, let’s just take 3. 3 to the 1 = 3. -3 squared is positive 9. -3 cube = -27.
-3 to the 4th is positive 81. -3 to the 5th, we talked about this a little in the last video. Three to the 5th is 243, so -3 to the 5th is -243 and -3 to the 6th is positive 729. So again, notice we have this alternating pattern, we solved this alternating pattern in the previous video.
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 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.
A Negative Fraction
Finally, the last case, a negative fraction. That is to say a negative base between -1 and 0. So this would be a number that is negative and has absolute value less than one. It is between -1 and 0. So let’s take -1/2. -1/2 to the 1 is of course -1/2.
-1 squared is positive 1/4. -1 cubed is -1/8. -1 to the 4th is positive 16. -1/2 to the 5th is -32. -1 to the 6 is positive 64. -1/2 to the 7th is negative 128.
And then positive 1 / 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 0 but we’re getting closer to 0 by jumping back and forth above 0 and below 0.
We’re approaching 0 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 of the 6? Well there’s no clear answer, it would be true for positive numbers greater than 1 and false for negatives.
Also, if x = 0, x to the 7th would l equal to x to the 6th which would be 0. And that’s 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 1 and x is unequal to 0, is x to the 7th greater than x to the 6? 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 then any negative. So therefore, we’re gonna get a no answer 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 0 and 1? So these are the only positive numbers allowed, the positive numbers between 0 and 1. Well, if we square say 2/3, square that we get 4/9, if we cube it, we get 8/27. Now notice 4/9, 2/3 is above 1/2, 4/9 is slightly below 1/2, it is above 1/4.
8/27, well 8/27 is definitely less than a 3rd. 4/9 is greater than a 3rd. 8/27 is less than a 3rd. Then we get to the 16/80 first, 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 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 of the 6th is gonna be bigger than x of the 7th. This is also gonna 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.