excellent.
Negative exponent m^-1/4 becomes a quotient where the negative sign becomes a numerator of positive 1 and the m^1/4 becomes the denominator.
The remaining 5n multiplies with the 1 numerator above the denominator.
The equation cannot be further simplified thus left as 5n/m^1/4.
My question now is, why does the negative sign becoming positive numerator? Is that just the rule? What is the explanation for doing so?
I guess i could ask the same thing as to why is a negative property become a quotient.. Anyways just trying to deepen my understanding of how it works.
This is a great question. I truly mean that.
Exponential notation started as an abbreviation for multiplication by the same number.
[math]a^2 = a * a;\ a^3 = a * a * a; \text { and } a^4 = a * a * a * a.[/math]
Simple idea. As has often happened a good notation suggested new ideas. Notation is important.
[math]a^4 * a^5 = (a * a * a * a)(a * a * a * a * a) = a * a * a * a * a * a * a * a * a = a^9 = a^{(4+5)}.[/math]
That leads to the thought that [imath]a^b * a^c = a^{(b+c)} \text { if } b > 1 < c.[/imath]
Then someone brilliant thought
[imath]a * a^3 = a * (a * a * a) = a * a * a * a = a^4 = a^{(1+3)} \implies a^1 = a.[/imath]
Then someone equally brilliant thought
[imath]1 * a^3 = 1 * a * a * a = a * a * a = a^3 = a^{(0+3)} \implies a^0 = 1.[/imath]
This leads to a whole new definition of what non-negative exponents mean
[imath]\text {If } n \text { is a non-negative integer, then } n = 0 \implies a^n = 1; \ n> 0 \implies a^n = a * a^{(n-1)}.[/imath]
In other words, mathematicians discovered the idea of exponents by following the logic of their notation.
So what about exponents that are negative integers, Following the logic of our notation, we want to get
[imath]a^b * a^{-b} = a^{(b-b)} = a^0 = 1 \implies a^{-b} =\dfrac{1}{a^b}.[/imath]
In other words, what are called the laws of exponents are really the consequences of some definitions that grew out of a simplified notation.
Morris Klein suggested decades ago that students would understand math better if we explained how it actually developed historically rather than explained it in the way that twentieth century mathematicians deduce it logically.
If you think about the laws of exponents as defintions to memorize, they become a whole lot easier. After a while you forget the definitional basis and historical development and just accept them as Platonic truisms. (I think Platonism is intellectual nonsense.)
