I know this may be silly but lately I've been wondering about powers of negative numbers and I came up with an ambiguous example: ( − 1 ) ( 22 10 ) I wanted to know if it was +1 or -1. My first attempt was writing it down as a root: ( − 1 ) 22 10 Minus one to the even power is obviously plus one, then 1 to the power of 1 10 is 1. Yet if we first reduce the fraction and then calculate we get a different answer: ( − 1 ) ( 22 10 ) = ( − 1 ) ( 11 5 ) = ( − 1 ) 11 5 Even though it's (I suppose) the same situation, ( − 1 ) 11 is and we end up with − 1 5 , a complex number. What is (or is there even) the value of ( − 1 ) ( 22 10 ) ?

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I know this may be silly but lately I&#039;ve been wondering about powers of negative numbers and I came up with an ambiguous example:  (  −  1      )          (              22        10            )      I wanted to know if it was +1 or -1. My first attempt was writing it down as a root:            (      −      1              )              22              10  Minus one to the even power is obviously plus one, then 1 to the power of       1    10   is 1. Yet if we first reduce the fraction and then calculate we get a different answer:  (  −  1      )          (              22        10            )        =  (  −  1      )          (              11        5            )        =            (      −      1              )              11              5  Even though it&#039;s (I suppose) the same situation,   (  −  1      )      11   is and we end up with             −      1        5  , a complex number. What is (or is there even) the value of   (  −  1      )          (              22        10            )      ?

Alec Blake · Accepted Answer

(              a        b            )        c    =      a    b        a    c   is not an equation that holds for all   a  ,  b  ,  c. Although that equation is true when all of   a  ,  b  ,  c are positive real numbers, as you&#039;ve discovered it&#039;s not true in general.

icedagecs · Accepted Answer

HINT:  −  1  =      e          i      π      Now,  (  −  1      )          22      10        =      e          i      π              11        5              =  cos  ⁡            11      π        5    +  i  sin  ⁡            11      π        5    =  −  cos  ⁡            2      π        5    −  i  sin  ⁡            2      π        5

I know this may be silly but lately I've been wondering about powers of negative numbers and I came

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