text: When  is odd, 's asymptotic behavior reverses from positive  to negative . For ,  will also tend towards positive infinity with increasing , but towards negative infinity with decreasing . All graphs from the family of odd power functions have the general shape of , flattening more in the middle as  increases and losing all flatness there in the straight line for . Functions with this kind of symmetry  are called odd functions.
  For , the opposite asymptotic behavior is true in each case.
  Table of powers of decimal digits
  Rational exponents
  If  is a nonnegative real number, and  is a positive integer,  or  denotes the unique positive real th root of , that is, the unique positive real number  such that
  If  is a positive real number, and  is a rational number, with  and  integers, then  is defined as
  The equality on the right may be derived by setting  and writing
  If  is a positive rational number, , by definition.
  All these definitions are required for extending the identity  to rational exponents.
  On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real th root, which is negative, if  is odd, and no real root if  is even. In the latter case, whichever complex th root one chooses for  the identity  cannot be satisfied. For example,
  See  and  for details on the way these problems may be handled.