Help me help a kid with math (or, justifying negative exponents)

[Mokele]

Ok, so on this other forum I go to, one of the users asked for help explaining exponents to her kid and how the work for his homework. Most of it I could cover, but one thing I couldn't.

 

We all know negative exponents, and simple stuff like 1/(x^-2) = x^2, and it's easy enough to simply state the rule, but my question is *why* is that rule the way it is? I mean, what's the derivation behind it? I hate to say "it just is that way, kid", but it's been a hell of a long time since I took Algebra, and I don't remember the reasoning for it.

 

So, basically, what I'm after is a logical explanation for the way negative exponents work, that I can present in a form a kid taking HS algebra could understand. To give an example, I explained why 2^3 * 2^2 = 2^5 by expanding it out into (2*2*2)*(2*2), removing the parenthesis, and showing that 2*2*2*2*2 can be colapsed into 2^5. That's the sort of level answer I'm looking for (though I can simplify if needed).

 

So, anyone know the justification for the rule for negative exponents?

 

Mokele

[Mokele]

nevermind, wikipedia came to my rescue

[Treadstone]

it has to do with maintaining continuity of notation. Same with numbers to the zero power equaling 1.

[YT2095]

this may be of NO HELP at all (and I hate maths too!), But.

 

the way one of my GOOD teachers taught me was imagine the signs + or - as male or female in a pub/bar

male + female - (or anyway you want to use them)

 

two males will get on just great so that`s a +

two females will get on, so that`s a + also

a male and female may argue in a pud so that`s always a -

 

I know it works for some maths stuff where the - is used, I beleive it applies equaly here too

 

edit: and if I rem correctly multiplying by a - is the same as a Divide by or something like that

 

where`s Dave or Mat when ya need em eh!? )

[Panic]

Let me try to help!

Forgive me in advance if I only confuse you more but I will try to explain as best as I can.

 

The reason you may be having a problem with it it’s because of the Negative sign. Right!

 

What it the direct relation with negative number as exponential.

 

Well it’s just convention and uniformity nothing more. Just the same way we 1e3 = 1000 or , or E = Voltage in some book and V = Voltage in other. Some one along time ago decided that x^-1 would = 1/x but the negative (-) only point out that it is the inverse of the power we are representing since x^1 = x, then 1/x = 1/(x^1) = x^-1

 

Here are a couple of examples that will help you out if we let * equals multiplication sign the way that exponentials behave for multiplication and division.

 

x*x*x*x*x = x^1* x^1* x^1* x^1* x^1=x^(1+1+1+1+1) = x^5

in the same manner

x*x*x*x*x = x^2*x^3 = x^(5-3) = x^2

 

This shows us that the relation ship between multiplication of exponentials with the same base x results in the base to the power of the sum of the exponents.

 

So the inverse: if we wanted to reduce the power we get the following.

if we wanted to get x^2 we could say

x^2= x*x*x*x*x / (x*x*x) which is equal to x^5 / x^3 also = x^(5-3) = x^2

you see now that the powers subtract. 3(x’s) from the top would cancel out 3(x’s) from the bottom leaving only x*x

 

x*x*x =x^3

x*x =x^2

x =x^1

x/x =x^0 = 1

x/(x*x) =x^-1 =1/x

and so on and so on and

 

I hope this helps and good luck in math! It’s a universal language.

[Panic]

[bUBBLE]

sorry i screwd up a line above but here is what i was trying to say

same as

x*x*x*x*x = x^2*x^3 = x^(2+3) = x^(5)

[/bUBBLE]

 

 

sorry!

[Treadstone]

panic has it right, its simply convention that maintains continuity in notation. THe eastiest example is this...

 

x^3/x^2 = x^(3-2) = x^1= x

 

x^3/x^4 = x^(3-4) = x^(-1)

 

so what is x^(-1)? well if you write it out on paper is cross out fators its a little easier to see, but this should give you the idea.

 

x^3/x^4 = (x*x*x)/(x*x*x*x) notice, three of the x's on top cancel out with 3 on bottom so that means that x^3/x^4 = (x*x*x)/(x*x*x*x) = 1/x

 

So, because of this, we adopt the notation x^(-n) = 1/x^n in order to maintain uniform notation.

[jcarlson]

also:

 

3^3 = 27

3^2 = 9 or 27/3

3^1 = 3 or 9/3

 

so it follows that if the pattern continues:

 

3^0 = should equal 3/3, or 1

3^-1 = 1/3

[bloodhound]

Its due to the uniqueness of inverses in the group. if you take the multiplicative group R*. 3 (1/3) = 1 which is the identity. there fore the multiplicative inverse of 3 which is denoted by 3^-1 = 1/3