A Treatise of Algebra. 5 j

other. But it is to be observed that, in resolving anyExpression in this Manner, the Factor out of which theRoot is to be extracted, is always to be taken the greatestthe Cafe will admit of. It also may be proper to takeNotice, that, this Kind of Reduction is chiefly useful inthe Additjon and Subtraction of surd Quantities, and inuniting the Terms of compound Expressions, which arecommensurable to each other, or where the irrationalPart (or Factor) after Reduction, is the same in eachTerm. _

Thus V s 18 + v^ 2 is reduced to }/z -j- ^/z or"j /2 ; and /8a*4" </5 oaZ — / 720? i s reduced to 2a/2-f- 5W2 — 6a/ 2, or a/ 2. Moreover, by Reduction,

iff- _ -f will become -f

5 4_ _ 20 _

or ^s/l'+ Vs/?,°t iwv/i?.20 20 20 20

Also ^a/^a 2 * -j- 8 >' + %x/9 a* -J* *8a*a* becomes

6ax 1 /a 1 -j- 2 2 -}- gax/a 2 -^- 2x 2 or i^ax/a 2 4- a* 2 .

And za^2yb 2 x et.~x~ t 4 " $*/%a b* — %a 1 b 2 x be-

- I -=^=r 3-

comes 6a x a — * x ^b 2 'V. a —■*4' 6ax^/ab 2 — b 2 x or

6a 2 /ab 2 — b 2 x ; and so of any other.

The Reason of the preceding Rule and the Operationsdepending thereon, is manifest from what is shewn inp. 1 3 . and p. 4!. »

Surd Quantities, under different radical Signs, are reducedto the fame radical Sign, by reducing their Indices to theleajl common Denominator.

Thus a 1 ' and a*', reduced to the same radical Sign,

will become a* and a 7 (for the Indices are here 7 and

j, and these are equivalent to ) and where both In-

dices or Fractions have the fame Denominator). In the

fame Manner P* and "zP will become pV<r and p s ,

-I i

or Plr and p 1 '. And, universally , A m and B ? , will,

£ 4 when