Cor. If p is also a positive integer, then
a^m a^n a^p = a^{m+n+p};
and so for any number of factors.
212. Prop. II. To prove that a^m \div a^n = a^{m -n }, when m and n are positive integers, and m is greater than n.
a^m \div a^n = a^m \div a^n = {a . a . a \ldots to m factors}{ a a . a \ldots to m factors}, = a . a . a \ldots to m-n factors = a^{m-n}.
213. Prop. III. To prove that (a^m)^n = a^{mn } when m and n are positive integers.
(a^m)^n = a^m . a^m . a^m . \ldots to n factors = (a . a . a \ldots to m factors) (a a . a \ldots to m factors) \ldots the bracket being repeated n times, = a . a . a \ldots to mn factors = a^{mn}.
214. These are the fundamental laws of combination of indices, and they are proved directly from a definition which is intelligible only on the supposition that the indices are positive and integral.
But it is found convenient to use fractional and negative indices, such as a^{{4}{5}}, a^{-7}, or, more generally, a^{{p}{q}}, a^{-n}; and these have at present no intelligible meaning. For the definition of a"^ [Art. 210], upon which we based the three propositions just proved, is no longer applicable when m is fractional or negative.
Now it is important that all indices, whether positive or negative, integral or fractional, should be governed by the same laws. We therefore determine meanings for symbols such as a^{{p}{q}}, a^{-n}, in the following way : we assume that they conform to the fundamental law, a^m a^n = a^{m +n} and accept
