Strange Facts about Exponents

Sometimes things just have to be proven.

We are taught to accept some things as fact, but they are not as evident as the facts I know intuitively.  These are the "math facts" that must be derived from the basic things.

This can be done using the "shorthand" known as the exponent.  It is like a "component" but set over to the side.  It is not part of the number but is a reminder that the number has been acted upon by multiplying it by itself.  The common exponents are the "square" [2], and the "cube" [3], drawn from the construction industry: flat panel and a "box."

When working with "whole" numbers, that is to say, zero and all the numbers we count with (aka "natural numbers), the exponent tell us at a glance that things just got a lot bigger.

Exponents also work with what are called "rational" numbers and even with 'irrational" numbers.  And lo, and behold, exponents work with "imaginary numbers" -- all these are things I know, but they need to be shown to be true.

Squares, Cubes and Beyond

First an illustration:

Consider that you are going to build a box, not quite as big as Noah's box (ark) but perhaps about the size of what that ancient Jewish guy, Bezaleel, did with some acacia wood (the box: the ark).  We'll skip the gold.  Anyway, the point is, measurements are made for the box that included its "footprint" (length and width) and its "body" (adding height).  The ends of the box were square, having the same height and width.

Given a width, call it W, then the board will be cut that wide in two (2) directions to get a square.  Or, in this case, W squared.

W x W = W^2

Well unlike Bezaleel or Noah, let us say the instructions to make the box the same measurements in all three dimensions -- put on your 3D glasses here.

Then, keep the measuring using the same mark on the stick, and you construct a cube!

W*W*W = W^3

No, not that "WWW" -- just a cube.  You know, like the description of the Holiest of Holies.  You don't know about that?  (leave a comment, I'll explain).

If you divide a cube by a square, you're back to a line the length of one of the edges.  So how does that help anybody?

Consider this algebraic express:

w^3/w^2 = W*W*W/W*W.

Going a step further:

W/W * W/W * W = 1 * 1 * W = W

OR

W^2/W^2 * W = W

Now, notice the relationship between the cube and the square:

W^3  = W^1.  W^1 = W^[3-2] or simply W.
W^2

The principle is, when dividing exponents, subtract those in the denominator from those in the numerator.

So, when when using the same base, the form N^D/N^D = N^[D-D] = N^0.  But wait, there is that zero again. N^0 = 1.  No matter what N represents!

All that to get to the first know fact of the day:

N^0 = 1.

Then, what happens if the exponent is larger on the bottom?

N^3   = N^[3-5] = N^[-2].
N^5

What in the world?

Simple, really. Look at it the "long" way:

      1*N*N*N     = 1/N*N.  The N^2 is in the denominator.
N*N*N*N*N*1

Think of it using the number 2.

2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1
2^[-1] = 1/2
2^[-2] = 1/4
2^[-3] = 1/8


So, what do I know?


N*D = multiplying N times D
But
N^D = multiplying by N, D times

N^0 = 1

N^[-D] =     1   
                 N^D

Coming Next: Making Arrangements