What exactly goes raising a number to a fraction mean?I apologise for asking something so fundamental, but what exactly does 2 2 5 actually mean? I get raising a whole number to another whole number x y means you are multiplying x with itself y times, but what does it mean when y is a fraction?

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What exactly goes raising a number to a fraction mean?I apologise for asking something so fundamental, but what exactly does      2          2      5      actually mean? I get raising a whole number to another whole number      x    y  means you are multiplying x with itself y times, but what does it mean when y is a fraction?

Jovan Wong · Accepted Answer

2                  2        5              =      2          2      ×              1        5              =            2              2              5  For any rational number   m  =      p    q           |       p  ,  q  ∈      N  ,          x    m   is interpreted as the       q          th       root of   x raised to the power of   p i.e.      x                  p        q              =            x      p        q

Craig Mendoza · Accepted Answer

Note the following law for integers       a          x      +      y        =      a    x    ⋅      a    y  . Now if we want to generalize this rule to fraction we note that  a  =      a          1        =      a                  1        2            +              1        2              =      a                  1        2              ⋅      a                  1        2            i.e.       a          1              /            2       is the number which multiplies with it self becomes   a or in other words       a          1              /            2        =      a  . In the same way       a          1              /            b       should be the   b:th root out of   a.Now for fractions we define       a                  b        c              =  (      a          b            )                  1        c             i.e. we take the   b:th power of   a and then take the   c:th root of that number. In your case      2          2              /            5        =  (      2          2            )          1              /            5        =      4          1              /            5      which is the 5:th root of 4 (a number which is irrational and thus not nicely expressed in other ways).

What exactly goes raising a number to a fraction mean? I apologise for asking something so fundamen

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