Exponents

1. Introduction to Exponents

An exponent is an elegant shorthand for multiplication.

Instead of 5 * 5 * 5, you can write 5³

Instead of 3 * 3 * 3 * 3 * 3 * 3 * 3, you can write 3⁷

The number that gets multiplied is called the base. The number of multiplications that occur is called the power. So, in the above example, 3 is the base and 7 is the power.

Of course, this method only applies when the power is a positive integer. Later on, I will discuss what it means when a power is 0, positive, or even a fraction.

So 4² = 4 * 4 = 16

And 4³ = 4 * 4 * 4 = 64

And 4¹ = 4 = 4

2. x and y notation

In mathematics, when we want to talk about "any", we use a letter such as x or y or z. For example, if we wanted to say that 1 to any power equals 1, we could write this as follows:

1^x = 1

Using x-and-y notation, we can create a definition for the positive exponents.

Definition 1: Positive Exponents

x^y means x multiplied with itself y times.

x is called the base

y is called the power

3. Multiplication of Exponents

Multiplying exponents of the same base can be determined based on the above definition.

4² * 4³ =
= (4 * 4) * (4 * 4 * 4)
= 4 * 4 * 4 * 4 * 4
= 4⁵

So, when exponents get multiplied, if they have the same base, you can add the powers and create a new exponent.

Here are some more examples:

5⁵ * 5¹⁰ = 5¹⁵

2¹⁰ * 2¹⁰⁰⁰ = 2¹⁰¹⁰

Of course, this does not work if two exponents have a different base.

In mathematics, a method such as this can be presented as a theorem. A theorem is any statement that can be derived from previous results.

In this case, we are able to prove a theorem regarding the method of adding the powers of the same base. Here's the theorem

Theorem 1: x^y * x^z = x^(y+z)

(1) We know that x^y = x multipled to itself y times and that x^z = x multipled to itself z times. (Definition of Positive Exponents).
(2) Multiplying all those x's, we have (y + z) x's multiplied together.
(3) Now x multiplied to itself (x + z) times = x^{(y + z)} by the Definition of Positive Exponents.

QED

QED is put at the end of a proof to show it is done. It is an abbreviation for a latin phrase that means basically that the proof is finished. It serves the same purpose in a proof as a period does in a sentence.

4. Division of Exponents

To talk about division, it is useful to introduce the following definition:

Definition 2: Division

a = b / c means a is equal to b divided by c.

a is refered to as the quotient.

b is refered to as the dividend.

c is refered to as the divisor.

Division with exponents of the same base can also be determined based on the definition for positive exponents:

4² / 4¹ =
= ( 4 * 4 ) / ( 4 ) =
= 16 / 4 = 4
= 4¹

To divide two exponents of the same base, you simply subtract the two powers.

Here are some examples:

5³ / 5¹ = 5²

4¹⁰ / 4⁵ = 4⁵

Now, what happens if we are dividing by a number greater than the top (in other words, where the divisor is greater than the dividend)? In this case, we are left with a fraction.

5¹ / 5³ = 1 / 5²

4⁵ / 4¹⁰ = 1 / 5⁵

This leads us to a third definition:

Definition 3: Negative Exponents

x^(-y) means that we have a fraction of 1 over x multiplied by itself y times.

Here are some examples.

5^-1 = 1 / 5

4^-3 = 1 / 4³

And what happens if the subtraction results in 0?

We can answer this with the following theorem:

Theorem 2: x⁰ = 1

(1) By basic arithemitic, we know that
x⁰ = x^{(1 - 1)}
(2) Since 1 - 1 = 1 + (-1), we can rewrite this as:
x^{(1 + -1)}
(3) Now x^{(1 + -1)} = x¹ * x^(-1) by Theorem 1.
(4) Now, x^(-1) = 1/x, by Definition 3.
(5) So, we are left with x * (1/x) = 1

QED

We can also introduce a corollary to this theorem. A corollary is a small proof that is derived directly from the logic of a theorem.

Corollary 2.1: x⁰ = 1 implies that x ≠ 0

(1) Now x⁰ = x^{(1 - 1)}
(2) Which means that x⁰ = x / x
(3) But this implies that x ≠ 0 since division by 0 is not allowed.

QED

Another way of saying this result is that 0⁰ just like 0/0 or even 1/0 is undefined.

We can summarize division of exponents with the following theorem.

Theorem 3: x^y / x^z = x^{(y - z)}

Case I: y = z

In this case x^y / x^z = 1 = x⁰ = x^{(y - z)}.

Case II: y > z

In division, we are able to cancel out all the common factors. Since y > z, we cancel out z factors from both dividend and divisor and we are left with x^(y-z).

Case III: y < z
Again, we cancel out common factors. Since z > y, we are left with a fraction of
1 / [x^(z-y)] which, by definition 3, equals x^(-(z-y)) = x^(y-z)

QED

5. Fractional Exponents

There is more that we can talk about. What about fractional exponents such as x^(1/2)?

It turns out that based on our definitions, corrolaries, and theorems, we are now ready to take on fractional exponent.

Let's start with 1/2.

We know that x^1/2 * x^1/2 = x^{(1/2 + 1/2)} by Theorem 1.
Now x^{(1/2 + 1/2)} = x⁽¹⁾ = x.
So x^1/2 is none other than the square root of x.

Let's start out by looking at a definition for what a root is.

Definition 4: an nth root of x is a number that multiplied n times equals x.

Sometimes, nth roots are whole numbers. The cube root of 27 is 3 since 3 * 3 * 3 = 27.

Likewise, the 4th root of 16 is 2.

1 is its own 5th root since 1 * 1 * 1 * 1 * 1 = 1.

This gives us our last theorem:

Theorem 4: x^1/n = the nth root of x

(1) x^1/n multiplied by itself n times equals x^{1/n + 1/n + 1/n + etc.}.
(2) Now 1/n + 1/n + etc. n times equals n/n which equals 1.
(3) Therefore x^1/n multipled by itself n times equals x¹
(4) And this is the very definition of an nth root.

QED