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mitodoteira Group TitleBest ResponseYou've already chosen the best response.0
\[a,b,c \in \mathbb{R}\]\[Prove: a^{b}a^{c}=a^{b+c}\]
 one year ago

Noura11 Group TitleBest ResponseYou've already chosen the best response.0
Wait ! a should be a positive real !
 one year ago

mitodoteira Group TitleBest ResponseYou've already chosen the best response.0
a>1 I forgot to say that. Facepalm.
 one year ago

Noura11 Group TitleBest ResponseYou've already chosen the best response.0
It is true for all a>0. We can say : \[\Large a^ba^c=e^{\ln a^b}e^{\ln a^c}\\ ~~~~~~~~~\Large =e^{b\ln a}e^{c\ln a}\\ ~~~~~~~~~\Large=e^{b\ln a+c\ln a}\\ ~~~~~~~~~~\Large=e^{(b+c)\ln a}\\ ~~~~~~~~~~\Large=e^{\ln a^{b+c}} \\ ~~~~~~~~~~\Large=a^{b+c}\]
 one year ago

mitodoteira Group TitleBest ResponseYou've already chosen the best response.0
I can't use logs. I'm doing analysis and I'm still proving the fundamentals of real powers from the field axioms.
 one year ago

Noura11 Group TitleBest ResponseYou've already chosen the best response.0
OK ! If b and c are natural integers you can prove it using induction !
 one year ago

mitodoteira Group TitleBest ResponseYou've already chosen the best response.0
b, c are just real.
 one year ago

Noura11 Group TitleBest ResponseYou've already chosen the best response.0
@mitodoteira My last reply is the 1st step of the proof !
 one year ago

sauravshakya Group TitleBest ResponseYou've already chosen the best response.0
I dont think we can use induction here... three variables
 one year ago

sauravshakya Group TitleBest ResponseYou've already chosen the best response.0
Isnt it the property??? I am not sure if it can be proven. example : we know a*b=b*a because its a property what how to Prove it.
 one year ago

mitodoteira Group TitleBest ResponseYou've already chosen the best response.0
No, it isn't a property.
 one year ago

swissgirl Group TitleBest ResponseYou've already chosen the best response.0
hmmmmm how do you define the exponential function?
 one year ago

swissgirl Group TitleBest ResponseYou've already chosen the best response.0
http://math.stackexchange.com/questions/435751/provingtheproductruleforexponentswiththesamebase Here is the link to this particular question I asked on MSE. Take a look at both proofs. The proof using Least Upper Bound is more analytical and a touch harder to follow but I think that is the proof you are looking for.
 one year ago
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