MATHEMATICS ALGEBRA. 



[SOLUTIONS TO THE KXKBCICTS. 



S. Required the cube root of 3 to fire or six places of 

 decimal*. 







1 



1 

 1 



2 

 1 



- 1 

 3 

 4 



34 



4 



3-8 

 4 



2 

 4-2 

 4 



424 



4 



428 

 4 



' 



4-32 







1 



C -229444 

 BM8 



_ 4 



6/2,3,8002 



4 320 



It is easy to foresee, that after the step marked 4 

 (Ex. 3) is reached, the work of the subsequent steps can 



have little or no influence upon the three leading figures 

 6 '23 of the forthcoming divisors ; so that regarding these 

 three figures as constant, and recognising the others, 

 8092, only fur the sake of what is carried from them, wo 

 may, as above, make sure of at least three true decimals 

 of the root, beyond the three already found, by common 

 division, provided wo reject a figure of the constant 

 divisor, 6 -23, at each stop, taking care to secure accuracy 

 in the carryings from the rejected figures. 



Ex. 1. The cube root of x" + 9~* 5 + Cx* OOi' 

 42x + 44Le 343 is ** -f 3r 7. 



2. The cube root of x + Cx* -f 40x 3 96* 64, 

 that is of x + 6x+0.c* 40x s +0.r s + 96z 01, 

 isx*+2.r 4. 



3. The cube root of 12994449551 is 235L 



4. The cube root of 2 is 1 -25992104989. 



5. The cube root of 959 is 9-8014218. 



0. The cube rootofz" loxty+G'Jj;*!/- 133.cV 

 GOxy* Si" is z? 5xy 2i/ a . 



We here conclude the treatise on Elementary Algebra. 

 The subject, in its widest acceptation, is one of very con- 

 siderable extent we might almost say of unlimited 

 extent ; as there are no definite bounds to its operations, 

 hi the preceding treatise, our object has been to imfuM 

 to you, fully and perspicuously, the leading principles of 

 the science ; and thus to lay a sufficiently secure basis for 

 future researches. There is one department of the 

 subject the general theory of Logarithms and Series 

 which we have not touched upon here. It is a part of 

 Algebra which is marked by peculiar features, and is 

 occupied with investigations different in kind and in 

 object from those necessary for the solution of au alge- 

 braical equation, or for the reduction of an algebraical 

 expression ; and is, moreover, of sufficient importance to 

 merit distinct consideration. We shall, therefore, first 

 give solutions to the various exercises contained in the 

 foregoing chapters ; and in the succeeding one, the 

 subject of Series and Logarithms will be fully en: 

 into. An extended Table of Logaritlmis of natural 

 numbers will be also included. 



SOLUTIONS TO THE EXERCISES IN THE CHAPTERS ON ALGEBRA. 



' AT page 452 of the ALGEBRA, a promise was given to 

 furnish the ANSWERS to all the Examples proposed for 

 exercise in that subject. Considering for whom the 

 work was expressly written for persons not merely 

 unacquainted with the very alphabet of Algebraical 

 nee, but also precluded from the advantages of 

 academical instruction we have thought that more 

 i acceptable service would be rendered by supplying 

 ; sketches of the solutions themselves, rather than a mere 

 i register of the results. You will therefore regard wh:il 

 follows as furnishing a KEY to the unworked examples ; 

 showing briefly but, we hope, clearly the processes by 

 which the answers are to be obtained. 



In Addition and Subtraction, however, all that can 

 hero be done is to put down the totals ; for the result of 

 an addition or subtraction example exhibits, in itself, the 

 whole work. 



EXERCISES. PAGE 452. 



1. 3o + 40 = 3.4 + 43 -12 + 12=24. 



2. 5c 2a-6.3 2.4 = 15 8 = 7. 



3. 13n + b 13 1 + 2 = 13 + 2 = 15. 



4. 21m 9d = 21.8 9.5 = 168 45 = 123. 



6. 7d + 4n 20 = 75 + 4 2.4 = 35 + 4 8 = 31. 

 0. 3o + 46 6c = 3.4 + 4.2 5.3=12 + 8 15 = 6. 



7. 6m 6n 36 = 6.8 5 3.2 = 48 6 6 = 37. 



8. 14 3e + m = 14 3.3 + 8 = 14 9 + 8 = 13. 



9. 116 + n 13 = 11.2 + 1 13 = 10. 



10. _4d + 5tn 2n = 4.5 + 5.8 2=18. 



1L -+6-- + 6--6 + 6-4 = 8. 



rt 5.4 . 4.5 

 - 2 = -g- + -g 2.3 = 10 +4 6 = 8. 



13 . ,^ + m _, l = *i+^ + 8 _ t = 2 + 8 



1 = 9. 

 3a 46 + 6c 2 3.4 4.2 + 6.3 2 



-Li. J ~ "^ .rv. 



10 





10 



= 2. 



15. Sab + dm 66 + Ocn, 18 = 3.4.2 + 5.8 5.2 

 + 6.3 18 = 54. 



16. 2abm 3cdm + 



17. 



JJ.6..8 



8 

 6am 26c + n 49 



+ 4.5 197. 



140 



+ ad = 2.4.2.8 3.3.5.8 + 



-197. 



6.4.82.2.3 + 149 



IK) 



24 , 6a6c_ 60^_mjt-_ 

 18> m" 1 " 24 bed 26+d = 8 



24 ,5.4.2.3 60 

 ~ i L'l ~~2.37& 



8 + 1 



2.2 + 5. 



'5. 



Page 453. Ex. 1. 2xy ixy + 13.ry + 3.ry= Ury. 



2. ajci 7at-2 + 1 1 ajcf + axz = 4arz. 



3. Dmnj + 5mnx 13mnx = mm. 



a , a a ' a a a 



