A ‘Treatise of Algebra. ig

by */ A x B, that is, by that Quantity which being mul-tiply’d into itself will produce the Quantity AB.

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In the same Manner the Product of y'A by fB will

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appear to be y'A x B or y'AB : For if y'A be denoted3 —

by * and t/B by y, or, which is the same, is A — xxx

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and B — yyy, then the Product of y'A by fB will bedenoted by xy, and it’s Cube by xy x xy x xy ; but xy Xxyy.xy being, by what is already proved, equal tomxyyy, it is manifest that the Cube of the Product willlikewise be expressed .by xxx x yyy or it’s Equal AxB,

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and therefore the Product itself by y'A x B, as was tobe shewn. The like of any other.

5 0 . The Produft of any two Powers or Roots of the fameQuantity, is expressed by that Power or Root-of the sameQuantity whose Exponent is equal to the Sum of the twogiven Exponents : But it is to be observed that the Expo-nents here understood, are those defined in p, 6 , whereRoots or Surds are represented as fractional 'Powers.Thus the Product of x3 by x s is x 8 ; that of a -J-~z^3 by

.. .. i —. g t 4 — A

a-\-zV is a-f-zl ; and that of x+ (or fx) by x is

x T ; also that df a 2 -j- z 2 }* by a 2 -f- z 2 l^is (a 2 -s-L 2 lH 3 ^or a z f z 2 l r , that is a 2 -(- z 2 : 'Moreover the Product ofC+ 7 I 1 by r+yl 3 is (r c -j-y' 6 ; and that

of x i by x^ is x 2 or x + , and so of any other.

The Reason of these Operations may be explain’d inthe following Manner. First, if the Exponents arewhole Numbers, as in Example 1 , the Demonstrationis evident from Lemma II, p. 13 , for, by what is thereproved, x3 into x s , or xxxxx into xxxxxxxxx isequal to xxxxxxxxxxxxxxx or x 8 ; and so ofany other. But if the Exponents are Fractions, or itwere required to multiply c -f-yT by.e -f-yl 1 as in Exam-ple 5 ; let c -f-yV be denoted by x; that' is, let the Quan-

C 2 tity