anonymous
  • anonymous
Prove: a^b*a^c=a^(b+c) for real a, b, c.
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
\[a,b,c \in \mathbb{R}\]\[Prove: a^{b}a^{c}=a^{b+c}\]
anonymous
  • anonymous
Wait ! a should be a positive real !
anonymous
  • anonymous
a>1 I forgot to say that. Facepalm.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
It is true for all a>0. We can say : \[\Large a^ba^c=e^{\ln a^b}e^{\ln a^c}\\ ~~~~~~~~~\Large =e^{b\ln a}e^{c\ln a}\\ ~~~~~~~~~\Large=e^{b\ln a+c\ln a}\\ ~~~~~~~~~~\Large=e^{(b+c)\ln a}\\ ~~~~~~~~~~\Large=e^{\ln a^{b+c}} \\ ~~~~~~~~~~\Large=a^{b+c}\]
anonymous
  • anonymous
I can't use logs. I'm doing analysis and I'm still proving the fundamentals of real powers from the field axioms.
anonymous
  • anonymous
OK ! If b and c are natural integers you can prove it using induction !
anonymous
  • anonymous
b, c are just real.
anonymous
  • anonymous
@mitodoteira My last reply is the 1st step of the proof !
anonymous
  • anonymous
How?
anonymous
  • anonymous
I dont think we can use induction here... three variables
anonymous
  • anonymous
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.
anonymous
  • anonymous
No, it isn't a property.
swissgirl
  • swissgirl
hmmmmm how do you define the exponential function?
swissgirl
  • swissgirl
http://math.stackexchange.com/questions/435751/proving-the-product-rule-for-exponents-with-the-same-base 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.

Looking for something else?

Not the answer you are looking for? Search for more explanations.