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
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.
You keep dividing by that number and it eventually goes into fractions. A quick way to think about it is this :
4^4 = 4 x 4 x 4 x 4 = 256
4^3 = 4 x 4 x 4 = 64 and also = 256 ÷ 4
4^2 = 4 x 4 = 16 and also = 64 ÷ 4
4^1 = 4 = 4 and also = 16 ÷ 4
4^0 = 1 and also = 4 ÷ 4
4^(-1) = 1/4^1 = 1/4 and also = 1 ÷ 4
4^(-2) = 1/4^2 = 1/16 and also = (1/4) ÷ 4
If I’m not mistaken you’ve gotten the rule wrong.
4^-s = 1/4^s not 1/4s
So exponential are the base multiplied by itself so 4^3 = 4 *4 *4
If you then reduce the power by 1 you are basically dividing by the base.
4^2 = 4 *4 which is the same as (4^3 ) /4 which equals 4 *4 *4 /4
So 4^1 = 4 therefore 4^0 = (4)/4 =1
If you keep reducing the power you get 4^-1 = (4^0 )/4 which is 1/4
Then 4^-2 = (4^-1 )/4 which equals (1/4)/4 or 1/4^2
……4^-s = 1/4^s
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