5.3 Counting Numbers as Exponents
For counting numbers n and m,
- x^mx^n=x^{m+n}
- (xy)^n=x^ny^n
In general, however, x^n +x^m \ne x^{m+n}
In the last section, we developed an exponential equation representing the population at time n and then asked some questions about it:
P(n) = 1000(1.02)^n
Before we answer the questions from the last section, namely, how to compute:
- P(0) --- this would represent the population at time 0
- P(-1) --- this would represent the population 1 year in the past
- P(1.5) --- this would be at time 1 year and 6 months.
Let's see how we can manipulate exponents in order to get a deeper feeling for how different exponential relate. An exponent is just a basic operation on a number much in the same way that multiplication, division, addition, and subtraction are basic operations on numbers. So, in order to become better acquainted with exponents, let's start by multiplying two exponentials and see what happens. Can we find some sort of pattern?
As an example, if we multiply 21 by 23, we get:
21 × 23 = 24 since 2 × 8 = 16.
Is there a general pattern to x^n x^m?
Try different counting numbers and see if you can find a pattern.
Explore!
If you do find the pattern, think about why it's true and then take a look at the discussion below.
To multiply exponentials with the same base, we just add the exponents.
Is this true for every possible base and every possible counting number? Let's use the power of abstraction to understand the specific example in a more general setting. In other words, let's call our base x, one of the exponents m and the other exponent n. Then,
\begin{matrix} x^mx^n=\underbrace{x\cdot x\cdot \ldots\cdot x} \underbrace{x\cdot x\cdot \ldots\cdot x}\\{\;\;\;\;\;\;\;\;\;\;\;\;m \; times \;\;\;\;\;\;n \; times\end{matrix}
But the right-hand side has a total of (m+n) x's. This tells us that:
x^mx^n=x^{m+n}
Once again, we used specific examples, noted the pattern and then generalized the result.
What if we have the same exponents but different bases? We found a pattern when we had the same bases, so it would be natural to play the same game with the exponents as well.
As an example, 32 × 42 = 122 since 9 × 16 = 144.
Try different counting numbers and see if you can find a pattern.
If you find a pattern, think about why it's true and then take a look at the discussion below.
Hopefully you noticed that when we have the same exponent, and we multiply two exponentials, the product consists of an exponential with the same exponent and a base that's the product of the individual bases.
Let's use the power of abstraction to see why this is true for any counting number. Let's call our exponent n, one of the bases x and the other one y. Then,
\begin{matrix} \;\;\;\;\;x^{n}y^{n}=\underbrace{x\cdot x\cdot \ldots\cdot x} \underbrace{y\cdot y\cdot \ldots\cdot y}\\{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n \; times \;\;\;\;\;\;n \; times\;\;\;\;\;\;\;\\=\underbrace{xy\cdot xy\cdot \ldots\cdot xy} \\{\; \; \; \; \; \; \; \;\;\;\;\;n \; times\;\;\;\;\ \\=(xy)^n\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{matrix}
But this tells us that:
(xy)^n=x^ny^n
Once again, we used specific examples, noted the pattern and then generalized the result.
What if we add exponentials with the same base and different exponents? We found a pattern when we multiplied exponentials with the same bases, so it would be natural to see if there's a pattern with addition, as well. Let's look at a specific example and see if we notice anything.
21 + 22 =
22 + 23=
23 + 24 =
Is there a pattern?
I'm afraid that here we have no recognizable pattern. When adding exponentials with different exponents, we can't combine them. In general,
x^m +x^n \ne x^{m+n}
Can we ever combine added exponentials? Yes, but only if they have the same base and the same exponent. As an example,
x^3+4x^3=5x^3
This is true, because x^3+4x^3 is equal to x^3(1+4) after factoring out the x^3term. Combining, we then get 5x^3; in other words, if we have the base and the same exponent, then we can factor and then combine.
Explore!
Combine the following into one expression using the exponent rules that you just learned. If none apply, then write "simplified"; use the "^" symbol for the exponent (for example, you can write x squared as x^2.
x^3x^6=
x^5+5x^5=
x^5+x^6=
x^2y^2=
Now that we have a sense of some of the basic rules of exponents, we can begin to think about exponents that aren't natural numbers. First, let's study an exponent of 0.
