Lesson · Exponents
Exponents
Multiplication is a shorthand for repeated addition. Exponents are the same trick one level up: a shorthand for repeated multiplication. This lesson builds the notation from scratch, then uses it to derive the first rule exponents follow.
What is an exponent?
One of the most powerful tools in mathematics is to eliminate complexity through notation. By abstracting away complexity, we allow ourselves to see the same problem from a whole new perspective, often revealing new insights that were hidden from us before. One of the very first examples a student comes across is the multiplication symbol:
Imagine a world where we didn't have it. What's
Did you have to slow down and count all the 7's? I did, and I'm writing this lesson! What if I instead asked you what 7 × 9 is? With just one simple symbol, ×, we make reasoning about repeated addition far easier on ourselves. In some ways, we allow ourselves to forget that multiplication is really just addition: we turn the trees into a forest.
In today's lesson, we will introduce exponents to encode the concept of repeated multiplication. Of course, multiplication is just repeated addition, so does that make exponents repeated, repeated addition? Repeated addition that's repeated? As you can see, from the perspective of exponents, the concept of addition gets quite challenging to reason about (addition that's repeated, repeatedly? That doesn't sound right either). Thankfully, since we can effectively forget that multiplication is repeated addition, we can easily say that exponents are repeated multiplication, in the same way that multiplication is just repeated addition.
The notation for exponents is a bit different from multiplication's × symbol, but it behaves in just the same way. To express
with exponents, we write 79. More generally, to express multiplying n 7s together, we write 7n:
When using exponent notation, we call 7 the base and n is the exponent.
Example: evaluating a power
To evaluate 24, we multiply together four copies of our base 2:
The base is 2 and the exponent is 4. Notice that 24 is not 2 × 4 = 8 — a common first mistake.
To evaluate 33, we multiply together three copies of our base 3, and for 62 we multiply our base 6 by itself:
Every power of 1 through 10
Now that we understand the basics, let's take a look at what happens when we raise the bases 1 through 10 to the powers of 1 through 5.
| base = b | b1 | b2 | b3 | b4 | b5 |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 4 | 8 | 16 | 32 |
| 3 | 3 | 9 | 27 | 81 | 243 |
| 4 | 4 | 16 | 64 | 256 | 1,024 |
| 5 | 5 | 25 | 125 | 625 | 3,125 |
| 6 | 6 | 36 | 216 | 1,296 | 7,776 |
| 7 | 7 | 49 | 343 | 2,401 | 16,807 |
| 8 | 8 | 64 | 512 | 4,096 | 32,768 |
| 9 | 9 | 81 | 729 | 6,561 | 59,049 |
| 10 | 10 | 100 | 1,000 | 10,000 | 100,000 |
Reviewing our table, we can easily read off the result of raising each base to each power. There are several points of interest to notice as we do so:
- The first row is all 1, since 1 × 1 = 1 and thus 1n = 1.
- The last row just adds a trailing 0 to each entry, with the number of 0s being the exponent on 10; e.g. 103 = 1,000, which has three 0s.
- The first column is just the base itself, since b1 = b for every base; e.g. 71 = 7.
- Unlike multiplication and addition, we quickly run into very large numbers with very small inputs; e.g. 94 = 6,561 versus 9 × 4 = 36 and 9 + 4 = 13.
The product rule
With our basic understanding in place, it's time to turn our attention to trying to understand what rules exponents follow. In order to do so, we will have to remind ourselves what exponents really are: repeated multiplication. As such, to figure out how exponents work, we will explore using the rules of multiplication. To start, let us return to our example of
If we were to calculate this by hand, showing each multiplication as a single step, then our next line would be:
and our next:
While we could continue in this manner, let us instead stop to notice 49 = 72 and 343 = 73, so these two lines are really just equal to:
and:
but wait, we can apply that same thinking to the first line since 71 = 7 and thus:
Now look closely at the right halves of these equations. Each one is nothing but a pile of 7s multiplied together — and that is exactly what a power of 7 is. So can we just swap in 78, 77 and 76 for them and be done?
We can, but it is worth slowing down for a moment to see why we are allowed to. Written out with no parentheses anywhere, a line like our second one is one long chain of multiplication:
Nothing in that chain tells us which multiplication to do first. We are used to working left to right, so we would multiply 49 × 7, then multiply that by the next 7, and so on. However, as we already know, multiplication is associative: for any three numbers, (a × b) × c = a × (b × c); for example:
Using the associative rule for our example, we can instead start multiplying the right end of 7s, leaving the 49 alone:
Doing the same substitution to each of our lines, we see that:
And if we continue in this manner, we also see:
As we can see, 79 = 7x × 7y when x + y = 9. This is not a special property of 7; if we performed this same exercise with any base, we would see the same result. More generally,
Ready to practice?
Take a quiz to practice this lesson three ways, and pick which of them you want before you start:
- What the words mean — name the base, name the exponent, and pick out what the lesson's terms stand for.
- Working out a power — read a power written out in words, like “what is 5 raised to the power of 3?”, and work out what it is worth.
- The product rule — evaluate a power whose exponent is a sum, given the value of each half.
Every answer is a power you can find in the table above, so you can always check your own work.
The PDF has all four practice sets, including the one a quiz can't ask for: evaluating a power by writing out every line of the repeated multiplication, which is an exercise in showing the working rather than in landing on a number.