Why Doesn't 2 1 / n = 1 / ( 2 n )Take 2 1 / n . Since 1 / n can be simplified as n − 1 , the original term can become 2 n − 1 . The exponents can then be multiplied to result in 2 − n which is 1 / ( 2 n ). However it is obvious that 2 1 / n ≠ 1 / ( 2 n )because the first item approaches 1 as n → ∞ but the second item approaches 0. What is wrong with the logic I used to simplify the first into the second that causes this discrepancy?

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Why Doesn&#039;t       2          1              /            n        =  1      /    (      2    n    )Take       2          1              /            n      . Since   1      /    n can be simplified as       n          −      1      , the original term can become       2                  n                  −          1                    . The exponents can then be multiplied to result in       2          −      n       which is   1      /    (      2    n    ). However it is obvious that      2          1              /            n        ≠  1      /    (      2    n    )because the first item approaches 1 as   n  →  ∞ but the second item approaches 0. What is wrong with the logic I used to simplify the first into the second that causes this discrepancy?

Josie123 · Accepted Answer

Exponents multiply in the sense that  (      a    b        )    c    =      a          b      c        .They do not multiply in the sense that      a          (              b        c            )        ≠      a          b      c        .

tr2os8x · Accepted Answer

The order in which you do the operations is vital here. Unlike addition and multiplication, exponentiation is not associative:      a          (              b                              c                              )        ≠            (              a        b            )        c  for &quot;most&quot; values of   a,   b and   cIt is true that       2          1              /            n        =      2                  n                              −            1                              , but that&#039;s because       2                  n                              −            1                               means       2          (              n                              −            1                              )      . As you saw for yourself,       2          (              n                              −            1                              )       is not the same thing at all as             (              2        n            )              −      1

Why Doesn't 2 <mrow class="MJX-TeXAtom-ORD"> 1 <mrow class="MJX-TeXAtom

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