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6 A 'Treatise os Algebra.

Thus the fifth Power of a, by placing 5 (the Indexof the fifth Power) above a, will be expressed by a S :Also the Cube, or third Power of b will be exprefl ed by£ 3 ; and the seventh Power of «+•*■, by a -|- xP ; allwhich, being mere Notation, does not require, nor in-deed admit of a Demonstration.

2°. But if the Quantity proposed be itself a Pouter orSurd , it utill be involved by multiplying it’s Exponent bythe Exponent of the proposed Power.

Thus the Cube, or third Power of a 1 is a 6 ; also thefifth Power of bl is b lS ; moreover the fourth Power of

lastly the third Power of

a — 1> i i is a —

The Demonstration of this Rule may be as follows.Let A” be taken to represent the proposed Quantity,('which may be any Quantity whatever) and let theIndex of the Power to which it is to he raised, be ex-pressed by m ; then, I say, the Power itself will be trulyexpresled by A" m (as by the Rule.) For since A" is thefame as A x A x A x A &c. continued to n Factors, thisraised to the Power, or multiply’d by itself m times,must consequently (by Lemma I and 2, p. 13.) be equal toAx Ax Ax AxAxAxA &c. continued to m times n Fac-tor, or to nm Factors ; which, by the Method of Nota-

tion, is the same with A'””, as was to be proved.

3 0 . A Quantity composed of several Factors multiply’dcontinually together , is involv'd by raising each Factor tothe Pouter proposed.

Thus the Square, or second Power of ab is a 2 b 2 * ; like-wise the third Power of 2 abc, is zlalblct or 8 a 3 i 3 c 3 -also the fifth Power of 3a x a 2 — x 2 xa-sbs-c is ex-pressed by 2+3 xa 2 — **1' x as- b-s- and lastly thethird Power of 2a 1 x F x a x}^ is expressed by a 2 xb

X a -)- r, f 3 and so in other Cafes.

But, in order to explain the Reason of these Opera-tions, let the Expression 2 a 2 xF x a -f- *1 T , in the lastExample, be again resumed. And, it is evident, from1 the