An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

Second Chapter.
Radicals and irrational numbers in general.
5. Repeated multiplication gives rise to a fifth operation, that of
raising to a power or Involution. () means a product which consists of n (exponent) factors, each equal to - (base). The values of
the powers of a positive base are only positive, those of a negative
base are positive or negative according as the exponent is an even
or odd number. Involution is always possible within the range of
rational numbers.
From one inversion of involution arises the problem of extracting
roots, Evolution. Given the positive number b where a and b are
relatively prime, it is required to determine a positive number x, so
that x =- b may be true for a given n. We assume the number -
positive, for, when it is negative and the exponent in even, as far as
our conception of number yet reaches, no such root can be extracted,
whereas when the exponent is odd, the nth root of the positive value
of is to be taken, but with a negative sign. In like manner, it is
to be remarked from the outset, that when the exponent is even, the root
of a positive quantity can be given a positive or a negative sign. Accordingly, the purpose of the following considerations is only to show
that the value itself can be determined.
If there be a fraction x =.. such that p - or bp- = aq, since
q q~ - -o bp' aq", since
a and b may always be assumed relatively prime and since we know
that there is only one way in which each number can be composed
of prime factors, we see that this equation must break up into a =-pn,
b == qf. Thus a positive fraction is only equal to the nt' power of a
positive fraction, when its numerator and its denominator are each
equal to the nth power of an integer; in particular, an integer can never
be equal to the Seth power of a fraction; for when b = 1, q must also
=1. To find out therefore whether - is the nth power of a rational
number, we must form the table of nth powers of integers and examine
whether a and b appear in it.