and therefore the definition of root (http://planetmath.org/NthRoot) gives the
Here, the exponent is an integer. For enabling the validity of (1) for the cases where does not divide we must set the following
The existence of the in the right hand side of (2) is proved here (http://planetmath.org/existenceofnthroot).
The defining equation (2) is independent on the form of the exponent : If , then we have , and because the mapping is injective in , the positive numbers and must be equal.
The presumption that is positive signifies that one can not identify all roots (http://planetmath.org/NthRoot) and the powers . For example, equals and , but one must not
The point is that is not defined in . Here we have and the mapping is not injective in . — Nevertheless, some people and books may use also for negative the equality and more generally where one then insists that
According to the preceding item, for the negative values of the derivative of odd roots (http://planetmath.org/NthRoot), e.g. , ought to be calculated as follows:
The result is similar as for positive ’s, although the root functions are not special cases of the power function.
|Date of creation||2014-09-21 12:12:39|
|Last modified on||2014-09-21 12:12:39|
|Last modified by||pahio (2872)|