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mitodoteira

  • 2 years ago

Prove: a^b*a^c=a^(b+c) for real a, b, c.

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  1. mitodoteira
    • 2 years ago
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    \[a,b,c \in \mathbb{R}\]\[Prove: a^{b}a^{c}=a^{b+c}\]

  2. Noura11
    • 2 years ago
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    Wait ! a should be a positive real !

  3. mitodoteira
    • 2 years ago
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    a>1 I forgot to say that. Facepalm.

  4. Noura11
    • 2 years ago
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    It is true for all a>0. We can say : \[\Large a^ba^c=e^{\ln a^b}e^{\ln a^c}\\ ~~~~~~~~~\Large =e^{b\ln a}e^{c\ln a}\\ ~~~~~~~~~\Large=e^{b\ln a+c\ln a}\\ ~~~~~~~~~~\Large=e^{(b+c)\ln a}\\ ~~~~~~~~~~\Large=e^{\ln a^{b+c}} \\ ~~~~~~~~~~\Large=a^{b+c}\]

  5. mitodoteira
    • 2 years ago
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    I can't use logs. I'm doing analysis and I'm still proving the fundamentals of real powers from the field axioms.

  6. Noura11
    • 2 years ago
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    OK ! If b and c are natural integers you can prove it using induction !

  7. mitodoteira
    • 2 years ago
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    b, c are just real.

  8. Noura11
    • 2 years ago
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    @mitodoteira My last reply is the 1st step of the proof !

  9. mitodoteira
    • 2 years ago
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    How?

  10. sauravshakya
    • 2 years ago
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    I dont think we can use induction here... three variables

  11. sauravshakya
    • 2 years ago
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    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.

  12. mitodoteira
    • 2 years ago
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    No, it isn't a property.

  13. swissgirl
    • 2 years ago
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    hmmmmm how do you define the exponential function?

  14. swissgirl
    • 2 years ago
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    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.

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