Rick Hecker

1100

Exponents are very simple things. They follow very simple rules. The rules which bound exponents are the very things which make exponents usable. As such, these rules can and must be accepted under all circumstances (unles we wish to write our own definitions, which would be a waste of time because we would end up with the same definitions that exist today).

The first rule of exponents is that any positive integer exponent means that one should multiply a variable by itself as many times as the exponent indicates. So:

x^2=x*x; x^3=x*x*x; x^7=x*x*x*x*x*x*x; ect

The second rule of exponents is that multiplied exponents require addition of powers. For example:

x^2*x^3=x^5; x^5*x^100=x^105; ect

Along those lines, dividing exponents requires subtracting powers.

x^105/x^100=x^5

The first part of our problem with exponents shows itself when we are asked to solve x^1.5

However, upon examination, we can resolve that 1.5 is merely 1+.5 (or 1/2). If we go back to our property of multiplication of exponents, we see that we can make x^1.5 into x^1*x^(1/2). This gives us x#

Similar examples can be approximated with any decimal. Each decimal must be broken into a fraction which can be turned into a root.

x^2.451 = x^2*x^(1/5)*x^(1/5)*x^(1/100)*x^(1/100)*x^(1/100)*x^(1/100)*x^(1/100)*x^(1/1000)

= x^2 *2 #*5#*#

To further this explination of exponents, we must see that x^(1/n) is the number, multiplied by itself n times to give the result x.

The second part of our problem lies in negative exponents. These, also, can be solved using our multiplication and division property. For instance:

x^-5 = x^4 / x^9 = x^(4-9)

So to find a numeric result for x^-5 we can find x^4 and x^9 and divide.

Negative decimal exponents work as a combination of the above rules:

x^-2.5 = x^(.5-3) = x^(1/2) / x^3

Once one sees how to seperate the exponents, it becomes easy to see the way to approximate decimal answers for seemingly complex problems. We can also see that calculators and computers could easily go through this same process of finding each part of the exponent and combining them to give the final result. For an example such as Pi, the computer would approximate 3.14...

After searching for hours on the internet, it was difficult to understand how no one was accredited with discovering negative and partial (decimal) exponents. At least that's what I thought until I discovered it myself. I believe the discoverer of these exponents is not widely published because the laws already existed which could compose these exponents. So, like all other mathematical progress, it was merely a discovery of a way to manipulate old rules to find new numbers. This, I believe, is the mathematical way of thinking: to take rules that already exist and create new rules from them, to solve new problems.