10. Each of the letters composing a term is called a dimension of the term, and the number of letters involved is called the degree of the term. Thus the product abc is said to be of three dimensions, or of the third degree; and axt is said to be of five dimensions, or of the fifth degree. A numerical coefficient is not counted. Thus 8a265 and a265 are each of seven dimensions, or of the seventh degree. 11. But it is sometimes useful to speak of the dimensions of an expression with regard to any one of the letters it involves. For instance, the expression 8a3l4c, which is of eight dimensions, may be said to be of three dimensions in a, of four dimensions in 6, and of one dimension in c. 12. A compound expression is said to be homogeneous when all its terms are of the same degree. Thus 8a6 – a462 +9ab5 is a homogeneous expression of six dimensions, or of the sixth degree. 13. In dealing with algebraical expressions, where the letters denote numerical quantities, we may make use of the principles with which the student is familiar in Arithmetic. Thus ab and ba each denote the product of the two quantities represented by the letters a and b, and have therefore the sarne value. Again, the expressions abc, acb, bac, bca, cab, cba have the same value, each denoting the product of the three quantities a, b, c. It is immaterial in what order the factors of a product are written; it is usual, however, to arrange them in alphabetical order. Example 1. If x=5, y=3, find the value of 4.x ́y3. 4x2y3 =4 x 52 x 33 =4x 25 x 27 8b.x2 Example 2. If a=4, b=9, xc=6, find the value of 27a3 • 8 x 9 x 36 =11. 14. If one factor of a product is equal to 0, the whole product must be equal to 0, whatever values the other factors may have. A factor 0 is sometimes called a zero factor." Example 1. If x=0, then abəxy2=0, whatever be the values of a, b, y. Example 2. If c=0, then ab%c3=0, whatever values a, b may have. EXAMPLES I. b. If a=7, b=2, c=0, x=5, y=3, find the value of 1. 4ax2. 2. a3b. 3. 8by. 4. 3.ry 5. 32.c. 6. 5b3y2 7. Exy4. 8. a'c. 9. a'cy. 10.82-By. 11. abr. 12. Jury". If a=2, b=3, c=1, p=0, q=4, r=6, find the value of 4ch2 15. 17. 3a2bc. 9a3 2ap 22. 3a2. 9a" 7r 5a"b? 27a? 64 23. 2a. 24. coba. 25. 26. 27. qno bor 28. zobo 86 992 64pa. 32 15. DEFINITION. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of 81 is 9, because 92=81. The square root of a is denoted by Ja, or more simply Ja. Similarly the cube, fourth, fifth, &c., root of any expression is that quantity whose third, fourth, fifth, &c., power is equal to the given expression. The roots are denoted by the symbols V, V, V, &c. 32=2; because 25 = 32. The symbol N is sometimes called the radical sign. Example 1. Find the value of 57(6a3b4c), when a=3, b=1, c=8. 5 (6ab4c)=5x (6 x 33 x 14 x 8) =5 x xv (6 x 27 x 8) If a=8, c=0, k=9, x=4, y=1, find the value of 1. (2a). 2. V(kx). 3. (2ax). 4. (2ako). 5. (3k). 6. Yax3). 7. \(82%y). 8. (cy). 9. 2x /(2ay). 10. 5y (4kx). 11. 3c 7 (kx). 12. 2xy (4y"). 25a 16x 13. 14. 15. 4943 ars 18. 162 (a2k24 19. 20. 3k 3x3 13. 7 ( 3 If a=4, b=1, c=2, d=9, x=5, y=8, find the value of 21. J(Sac). 22. 6 (463). 23. 7/(5dx). 1 24. Jy). 25, 26. 5xy? 8acd) 1 1 29. 4023 568 32. 7(8ab2c). 33. Jde. 34. Jya. 35. 164 36. d. 1 (sacd). 3 3 16. In working examples the student should pay attention to the following hints : 1. It should be clearly brought out how each step follows from the one before it; for this purpose short verbal explanations are often necessary. 2. The sign “=" should never be used except to connect quantities which are equal. Beginners should be particularly careful not to employ the sign of equality in any vague and inexact sense. 3. Unless the expressions are very short the signs of equality in the several steps of the work should be placed one under the other. 4. In all work too much importance cannot be attached to neatness of style and arrangement. The beginner should remember that neatness is in itself conducive to accuracy. Example 1. Find the value of gab - 7x2 – Jaya +263, when a=5, b=4, x=3, y=2. = 6 - 63 – 45 + 128 Example 2. Find the value of $.x2 – aạy + 7abx – 313, when a=5,b=0, x=7, y=1. =293 – 25 – 24 =116. NOTE. In the last example the zero term does not affect the result. Example 3. When p=9, r=6, k =4, find the value of pr 212 9k 2 x 36 + V (5p+3r+1) – +v(45+18+1) 9k 3 9x4 = 27 +64 - 2 = 4 x 3 +8-2 =63 EXAMPLES I. d. If a=2, b=3, c=1, d=0, find the numerical value of 1. 6a +56 - 8c+9d. 2. 3a-46+6c+5d. 3. 5a +36-26+6d. 4. ab + bc+ca – da. 5. 6ab – 3cd +2da - 5cb + 2db. 6. abc+bcd+cda+dab. 7. 3abc – 2bcd + 2cda – 4dab. 8. 2bc+3cd – 4da +5ab. 9. 3bcd+5cda – 7dab + abc. 10. ap+62 +62+d?. 11. 2a2+363 — 404. 12. a4 +64 – 04. If a=1, b=2, c=3, d=0, find the numerical value of 13. a3 + b3 + c3 +d3. 14. 03-a3-33 - abc. 15. 3abc - 62c – 6a?. 16. 2a2 +262 +2c2+2d2 – 2bc – 2cd – 2da – 2ab. 17. 23 + fad' - 32+ b2d. 18. a +262 +2c2+d2 + 2ab +2bc+cd. 19. 2c2 + 2a +262 – 4cb+Cabcd. 20. 13a2 + 1/4+20ab – 16ac – 16bc. 21. 6ab-aco - 2a+3/4 - 3d + $". 22. a? - c2 +62 - d2+ 2ab – 2cd. 23. 2ab - 363+3ac— 2c-d+1 ad. 24. 12574c-9d5 + 3abcd. If a=8, b=6, c=1, x=9, y=4, find the value of 32 Ga 25. Şa-123+ {y. 26.ax ycxy 3a26 6cyl 27. (323 28. 463 8x2 (by a 29. gabe- V - bey. -16)+ ( ) 32. |
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