M ATII KM ATICS. ALGEBRA. 



[EXTRACTIOX OF BOOTS. 



poMd number as w please ; we con thui approximate, 

 M it U called, to the square root of 3, or 6, or 7, <vc-> to 

 any degree of mmmoM. M WM ahown in the operation 

 for the square root in decimal arithmetic. There is 

 thus a marked difference between V 3 and J 3 ; the 

 value of the former can be approximated to, to any 

 of Murnfm : the raloe of the latter cannot be 



OTVUW VB MMMM HI II - _ 



approaehed to at all it hat no existence. It may lie 

 noticed, bowerer, that both turds and imaginary MM* 

 ties may sometime* be simplified in appearance. Thus, 



medied by marking the extent to which V reaches by a 

 Ur-vinculum, thus, 3^/ab 3 . x; but superfluous marks 

 and signs are always to be avoided, so that tho form 

 &r\/ab* U to be preferred ; a bar over ob is not re- 

 quired, when it U understood that J covers, or extends 

 to, all the factors which follow it We have, indeed, put 

 a bar over similar expressions at page 455, but did so for 

 Fear that you should limit tho influence of the radical ; 

 it is best omitted when there is no danger of such mis- 

 take being made. 

 1 1 



1 / <i *^ /^ 1 Ot* ' 

 the number under J can be separated into two factors, 

 one of which is a square ; the operation implied in J 

 being actually performed on this square, it becomes freed 

 from the radical, which then applies only to the other 

 factor. You should never leave under the radical, in 

 any final result, a factor upon which tho operation indi- 

 cated by the radical can INJ actually performed. Thus, 

 such an expression as */ a-b should be reduced to the 

 simpler form a J b. In like manner, x/a 3 * 3 !/ = ax V 

 y ; ^/16az* V 8'2 ox 8 2x V 2a ; and so on. It is 

 customary to call such forms as J b, */ y, S/ 2a, <fcc., 

 in which the operation under the radical cannot be per- 

 formed, otoebraicoi turd* ; although, if the letters were 

 interpreted, it is quite possible that the algebraical surd 

 might prove a rational number. Thus, if 6 were 4, then 

 J b would be -f 2 ; which is rational, though ambiguous 



as to sign. 



We shall now give you a few examples on the ttoowaoit 

 of simple quantities, or quantities consisting of but one 

 term. It has been stated (p. 461) that quantities con- 

 sisting of <ux> terms are called binotniaU; you will bo 

 prepared to expect, therefore, that those of on term 

 are often called monomial*, those of three terms trino- 

 mials, those of four quadrinomiaU ; a quantity consisting 

 of more terms comes under the general name of poly- 

 nomial or muttinomiot. 



To extract a proposed root of a simple or monomial 

 quantity. 



RULE 1. Write tho root of the coefficient with its 

 proper sign, remembering that an odd root, like an odil 

 power of any quantity, has the same sign as that quan- 

 tity ; but that an even root of a positive quantity takes 

 the double sign +. 



2. Divide the exponent of each letter by the exponent 

 of the radical ; or, which is tho same thing, multiply by 

 tho fractional index, by which the radical may be re- 

 placed ; and tho proposed root will be obtained. 



1. ^ 16o*x 4a a V *> . or 4a 2 *J. Here the square 

 root of the coefficient 10 is 4 ; the letters with their 

 exponents are a*z l , the index or exp. of the radical is 

 2 ; /. dividing the exponents by this, the letters of the 

 result are ajj-J a a zt. Or replacing the radical by 

 the equivalent exponent, the ex. is (16a'x)J = 4(a 4 x')} ; 

 and multiplying the exponents within the vinculum, by 

 the exp. without, we have 4(o'z')i = 4a'z^ 4o J x. 

 The double sign is not placed before tho 4 : tho reason 

 in, tliat tho extraction is not completed ; J x u & still 

 unperformed operation, and the double sign is regarded 

 as still implied in *.'. 



2. V 9o*x - 3a|xf - + 3az 



4] sy Sa"x s 2aJT- a^x 



It is to bo observed, that when on irrational or surd 

 quantity occurs among the factors of a term, the 

 composing that term are not arranged in alphabetical 

 order except when tho surd is expressed by a fractional 

 exponent ; when it U denoted by tho sign N ', it is always 

 placed last, to prevent all mistake as to tho extent of 

 influence of thin sign. If the result in tho last ex. h;nl 

 | been written 3^/ab'x, it would evidently have con 

 ft wrong meaning ; it is true this might have been re- 



The first of these forms, namely 



is that which 



would, in general, be employed; negative indices, though 

 of much importance in certain general investigations, 

 being seldom used in particular expressions. But we 

 must tell you that algebraists do not like to leave surd 

 quantities in the denominators of fractions ; for as has 

 been said before (page 400), they wish their final results 

 to be in a form tho best suited for arithmetical computa- 



tion. Now, suppose you hod to computep, a being 



interpreted to you ; say a = 11. You would have first 

 to find the square root of 11, to several places of deci- 

 mals; suppose four places were considered sufficient, 



you would then b^e-J-.-^j-.^, and the d 



,, here indicated would have to be performed. But 

 instead of taking this course, let numerator and deno- 



minator of -i- be multiplied by V a, which, from tho 



v* 

 first principles of arithmetical fractions, you know to be 



allowable, -A- would thus be changed into -Sii ; and 



. v V 11 3 ' 3166 Vmi 



you would have to compute -^=^-=jj * 



can note, in a moment, pronounce the final result to bo 

 3015 . . . , which, of course, is the same as the result of 

 , but it would have taken you much longer to 



discover it. Ex. 6 therefore should not be left in tho 

 state in which we have left it above ; the steps should 



1 1 _,/_ N'' 



Va 3 5 = ab* V" 



XXAUFLBS FOR EXERCISE. 

 2. 



5. J'< . 



9. 



4. y 



8. 



'? 



10. 



. >., 



13. ( Sa^V) !* (% 4 *V)~ 15. 



10. V ICa'xV" 17. (32<^x*v*)"- 18- (32cr s x-V)~i 



FRACTIONS. 



Algebraic fractions are treated exactly tho samo as 

 numerical fractions in common arithmetic ; the only 

 diflercnce being in the symbols, and not in tho operations 



To reduce a mixed quantity to an improper fraction. 



A quantity is called a mixed quantity when it is partly 

 integral and partly fractional ; such a quantity may 

 always bo changed into an equivalent one, wholly frac- 



| tioual in form ; the result is called an impmprr fniction, 



| as well in arithmetic as in al^rlu-a, Iwcause an integral 

 quantity is rwlly absorbed into it. The rule for reduc- 



i ing a quantity partly integral ami partly fractional to a 



j form wholly fractional, is ao follows : 



