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Math Help - Principle of mathematical induction

  1. #1
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    Principle of mathematical induction

    Use the principle of mathematical induction to prove that if  a \in R then  ( a^{m} ) ^n = a^{mn} , \forall m,n \in N


    TRUE OR NOT ?? :
    1) WANT TRUE FOR n=1 AND m=1 :
      (a^{1 )^ 1 } = a^{1.1} \to a=a
    2) SUPPOSE TRUE FOR n=k AND m=v :
     (a^ {v ) ^k }= a^{vk}
    3) WANT TRUE FOR n=k+1 AND m=v+1 :
     (a^ {v+1 ) ^k+1 }= a^{(v+1)(k+1)}  HOW I PROVE IT ???
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by flower3 View Post
    Use the principle of mathematical induction to prove that if  a \in R then  ( a^{m} ) ^n = a^{mn} , \forall m,n \in N


    TRUE OR NOT ?? :
    1) WANT TRUE FOR n=1 AND m=1 :
      (a^{1 )^ 1 } = a^{1.1} \to a=a
    2) SUPPOSE TRUE FOR n=k AND m=v :
     (a^ {v ) ^k }= a^{vk}
    3) WANT TRUE FOR n=k+1 AND m=v+1 :
     (a^ {v+1 ) ^k+1 }= a^{(v+1)(k+1)}  HOW I PROVE IT ???
    (a^{v+1})^{k+1}=(a^v\cdot a)^{k+1}=(a^v)^{k+1}\cdot a^{k+1}=(a^v)^k\cdot a^v\cdot a^{k+1}

    Spoiler:
    Since from the inductive step (a^v)^k=a^{vk},

    (a^v)^k\cdot a^v\cdot a^{k+1}=a^{vk}\cdot a^v\cdot a^{k+1}=a^{vk+v+k+1}=a^{(v+1)(k+1)}

    QED
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