Why is it that e.g. 4ˆ-s = 1/4 s?

Question

Hi! I hope my question is understandable. This is something I never quite understood.

Why is it that when you put the negative unit in the exponent that it can be translated into a fraction?

(I hope this makes sense! English is not my first language!)

Thanks in advance for the answers!

In: Mathematics

Answer 1

Basically the same as the other answers but perhaps a slightly different persepctive. We essentially just define x^(-n) = 1/x^n so that the rules of exponentiation still hold. Namely we know that if n and m are positive integers

then x^(n+m)=x^(n)*x^(m) and so we just define negative exponentiation so this relation continues to hold ie. 1=x^(0)

=x^(n-n)=x^(n)*x^(-n) implies x^(-n)=1/x^n.