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Math Help - Which number is bigger

  1. #1
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    Which number is bigger

    Hey guys, i need to prove without using calculator that:

    17 ^{14} >31^{11}
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  2. #2
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    Quote Originally Posted by mat1990 View Post
    Hey guys, i need to prove without using calculator that:

    17 ^{14} >31^{11}
    Notice that both 17 and 31 are "almost" powers of two, these are 16 and 32, respectively.

    So 17^{14}>(2^4)^{14}=2^{56}, and 2^{55}=(2^5)^{11}>31^{11}. With this, you find the inequality....
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  3. #3
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    Quote Originally Posted by mat1990 View Post
    Hey guys, i need to prove without using calculator that:

    17 ^{14} >31^{11}
    Alternatively, if

    16^{14}>32^{11}\Rightarrow\ 17^{14}>31^{11}

    16^{14}=16^3\left[16^{11}\right]=\left[2^4\right]^3\left[16^{11}\right]=2^{12}\left[16^{11}\right]

    32^{11}=2^{11}\left[16^{11}\right]

    2^{12}\left[16^{11}\right]>2^{11}\left[16^{11}\right]\Rightarrow\ 16^{14}>32^{11}\Rightarrow\ 17^{14}>31^{11}
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