Exponents, the Rules That Will Give You a Higher Power over Numbers
Episode #7 of the course Foundations of mathematics by John Robin
Are you ready to dig into the next topic of our foundations of mathematics course? Today, we will explore the wonderful world of exponents.
In Lesson 3, I showed you how to use exponents when we were learning to write out the product of primes in any composite number. For example, I mentioned that 288 is 2x2x2x2x2x3x3, but a more compact way of writing that would be 25(32). 1024 is 2x2x2x2x2x2x2x2x2x2. 2 is multiplied ten times. So in exponent notation, we would simply write:
1024 = 210
Much easier!
In this exponent, the number 2 is called the base and the number 10 the exponent. Think of the base as the number that’s being repeatedly multiplied together. It’s your starting point, aka home base.
Now, just like we had some rules with two square roots, we have rules with what happens when two exponents come together.
Let’s work with the number 2 as our example (but note that you can use any number).
1. If two identical bases are multiplied together, combine them and add their exponents:
This is the same as saying:
23 x 28 = 211
Think about this from the perspective of where exponents come from. 23 is 2x2x2. 28 is 2x2x2x2x2x2x2x2. How many total 2s (bases) are there? 3 + 8 = 11.
2. If two identical base are divided by each other, combine them and subtract their exponents:
This is the same as saying:
(28) / (23) = 25
Think about this from the perspective of where exponents come from. 28 is 2x2x2x2x2x2x2x2. 23 is 2x2x2.
What does this look like in a fraction?
(2x2x2x2x2x2x2x2) / (2x2x2)
We can cross out a 2 in the numerator for every 2 in the denominator, leaving us with 2x2x2x2x2, i.e., 25. This is the same as subtracting the number of 2s in the denominator (3) from the number of 2s in the numerator (8).
3. If a base to an exponent is itself the base of another number to an exponent, multiply the exponents together:
This one is a bit more complicated. It looks like this:
(23)8 = 224
Here, there are two bases. The first one, inside, is 2. It’s raised to exponent 3. The second one is (23) itself. It acts all as one base, raised to exponent 8.
To understand what’s happening here, let’s again look at it in terms of what the exponents are doing:
23 = 2x2x2.
So, we have:
(2x2x2)8
The exponent 8 tells us to multiply (2x2x2) by itself eight times. This looks like:
(2x2x2)x(2x2x2)x(2x2x2)x(2x2x2)x(2x2x2)x(2x2x2)x(2x2x2)x(2x2x2)
Does that hurt your eyes? Look at it only long enough to convince yourself that (23)8 is 224 because 23 is being multiplied by itself eight times, which is the same as multiplying 2 by itself 24 (3×8) times.
4. A negative exponent flips a fraction around, then becomes positive:
Simply put:
2-1 = 1/2
(1/2)-1 = 2
(3/59)-1 = (59/3)
If your exponent is bigger than 1, you simply apply rules one through three above on what remains:
(23) / (28) = 2-3 = (1/2)3= (1/8)
5. In a fraction exponent, the number in the denominator is a root:
We have worked a lot with square roots so far: √7, √2, √3. As an exponent, the square root has the exponent 1/2:
√7 = (7)1/2
Think in terms of what we’ve learned about exponents so far. The exponent tells us how many times to multiply a number by itself. In the case of a square root, we are multiplying it by itself 1/2 times. Doing this twice will give us the number back.
Using this idea, we can see, for example, that a cube root written as 3√7 would be three equal numbers that multiplied all together make 7. In exponent notation, this would be (7)1/3 and we can think of this as 7 multiplied by itself 1/3 times.
We can extend this to any root, n√7 (I’m using 7 as an example, but you can put any base in there). These are called nth roots.
If you have a fraction exponent with a numerator greater than 1, you simply apply rules one through four above:
163/4 = (4√16)3= (2)3 = 8
And that’s a look at what we can do with exponents. Stay tuned for tomorrow’s lesson, where we will explore their close relative: logarithms.
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