Monday, May 02, 2005

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:

1x = 1

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

Definition 1: Positive Exponents
xy 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: xy * xz = x(y+z)
(1) We know that xy = x multipled to itself y times and that xz = 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: xy / xz = x(y - z)
Case I: y = z

In this case xy / xz = 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 x1/2 * x1/2 = x(1/2 + 1/2) by Theorem 1.
Now x(1/2 + 1/2) = x(1) = x.
So x1/2 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: x1/n = the nth root of x

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

QED


4 comments :

Anonymous said...

Excellent

stuvxz said...

indeed
excellent!

Anonymous said...

Four to the fifth power divided by four to the tenth power is one over four, (with the four to the fifth power)- right?

Larry Freeman said...

Four to the fifth power = 4^5 = 4*4*4*4*4

Four to the tenth power = 4^10 = 4*4*4*4*4*4*4*4*4*4

Four to the fifth power divided by four to the tenth power = 4^5/4^10 = 4^(5-10) = 4^(-5) = 1/(4^5) = 1/(4*4*4*4*4)

Four to the fifth power divided by four to the sixth power = 4^5/4^6 = 4^(5-6) = 4^(-1) = 1/(4^1) = 1/4.

I hope that helps.