from the fact they express respectively the area of a square whose side is a, and the volume of cube whose edge is a. 17. The square root of a quantity is that quantity which, when raised to the second power or squared, will give the original quantity. It is generally written N. Thus, Vī6 = 4, V144 = 12. The cube root of a quantity is that quantity which, when raised to the third power or cubed, will give the original quantity. It is generally written 7. Thus, 38 = 2, 927 = 3, 71728 = 12. And so the fourth, fifth, &c., roots are indicated by the symbols V, V, &c., respectively. 18. The dimensions of an algebraical quantity are the sum of the indices or exponents of the literal factors. Thus, the dimensions of 3 a2b3c4 = 2 + 3 + 4 = 9, and the dimensions of aba“ = 1 + 1 + 2 = 4. 19. A homogeneous expression is one in which the dimensions of every term are the same. Thus, a3 + 3 aạb + 3 ab2 + 63 is homogeneous, Ex. II. 4. 6 {2 a3 – 4 (2 63 - 2 c3 - d)}; a’b + ab? + a'c + aca + b c + bc”. If a = 1, b = - 2, c = - 3, d = 0, e = 4, 5. (a + b + c + d + e)”; (a“ + 2 ab + 69 – c^) = (a + b + c). at - d4 c3 + 3 c d + 3 cdo + d3 " ad + ałd + ad + d3: 63 – 3 b'c + 3 bca – 63 7. Ve + € - 15 a2 + 62 ab + ac + bc bei 9c3 + 3 c + 3 C + 10 8. 3 (46 + 5 c)2 + 4 (c +e)? ; abcde. If x = 3, y = 4, % = 0, Find the value of 9. (3x – Vaca + y)2 (2 x + 3x + zja + 2). 10. {5 .** + 2 (y + 2)} {5 23 – 2 (y + 2)"}. 11. ad + you + me 12. (203 – ) = {3 202 + 3 (3 x2 + 3 xy + yo) y}. CHAPTER II. ADDITION SUBTRACTION, MULTIPLICATION, AND DIVISION. Addition. 20. RULE.--Arrange the terms of the given quantities so that like quantities may be under each other; add separately the positive and negative coefficients of each column; take the difference and prefix the sign of the greater, and annex the common letter. (When the coefficients are all positive or all negative, we, of course, simply add them together and prefix the common sign for the coefficient of the sum.) Ex. 1. Ex, 2. 3 a + 56 - 30 + 7 6 + C - 4d 2a - 7b + 40 2a - 2 b + 50 + 3d 5 at 6 - 20 7 a - 3 6 + 2c - 60 4a - 3 b + 8 C 2 a + 5 6 - 80 + 6d Ans. 14 a - 46 + 70 Ans. – 6 a + 7 6 - d Ex. 3. Add together 5 - 3 ya + 3 y, 6 ya + 7 xy – 4 x, 4 xy + 6 x – 5, - 2 22 – 3 xy + 2. Arranging like quantities in each expression under each other, we have :5 x2 - 3 y2 + 3 y :- 5 co'ns er Ex. III. Add together 1. 3 a – 2 b, 4 a + 7 b, 2 a + 3b, a – 5 b. 2. 9 a2 + 76%, - 3 ao + 4b, a’ + 6%, 4 ao – 12 . 3. a + b + c, 3 a + 2b + 3c, – 4 a + 76 – C, 2 b + 5 c. 4. X - Y - %, y – X – 2,2 – X – y, x + y + z. 5. 3 ao – 4 ab + 6 6, 7 ab - a– 6, 2 ao – 3 ab – 46, 4 ao + ab – 6. 6. 2 x4 – 7 x2 + 3, – 4 x 3 + 6 ac? – 2 x + 7, ** – 2 203 – 4 x, 6 23 – 9 – 12. 7. 2 ao + 7 ab + 362 – 6a - 56 - 2, ao + 3 a – 27 + 9, 9 ab – 2 a - 36 + 4, – 3 ao – 12 ab – 362 + 5a + 106 – 15. 8. 23 - x^y – Xox + xy” – xyz + xz, aʻy – wyü – xyz + yj— yoz + yz, xʻz – xyz – xz2 + yoz – yz + | 9. c + cos^ + cou, - dog – Royo – cao, y + c + c . 10. a? + ab? + aca + 2 ab – 2 aục – 2 abc, a’b + b3 + bco + 2 ab? – 2 abc – 202c, aʻc + 6°c + C + 2 abc – 2 ac – 2 bc?. 11. 2* – wy3 + wcz2 – 3 x*y + 3 xz, 3 x*y* + 3 ans + 3 xyoz, - 3 xyz - 6 x*yz, y* - acoy – ym + 3 xʻyj" – 3 xʻyz, – 3 xyz - 3 cao – 3 x + 3 ^^ + 6 cuoc, số + c^2 - 2x - 3 coga + 3 xʻz, 3 wyʻz + 3 xz3 + 3 yʻz* - 3 ym3 – 6 xyz-. 12. a* - ab + 3 aạca +. abʼc – 3 abc – 6%c, aob - a'c + 3 abc – 3 ac: – 6°C~ + bc, a'c – a'd + 3 ac – 3 acʻd + 6*c* - b'cd, - a* + ad – 3 aạc + 3 acʻd – ab'c + bạcd. Subtraction. 21. We have seen, Art. 7 (4.), that the subtraction of a quantity is equivalent to the addition of the same quantity with its sign of affection reversed. We therefore have the following rule : RULE. - Change the sign of each term of the subtrahend, and proceed as in addition. Ex. 1. Ex. 2. 5 x + 6y – 3% 6 a2 - 3 ab + 462 2 x – 2y + 2 - 2 ao – 3 ab – 2 62 Ans. 3 x + 8y - 5%. Ans. .8 a + 6 62. Ex. 3. and + 3 xạy + 3 xyo + 2 73 Ex. IV. 5. Subtract the sum of the quantities a* + 2 aʼba + 64, at – 2 aạb2 + 64 from 6 a* + 8 a2b2 + 6 64. 6. From 23 + 73 + m2 — 3 xyz take 4 203 + y2 + 4x3 + 3 xʻz + 3 xz2 – 3 xyz. 7. From 3 c4 + 3 ax: – 9 aʻxca + aʼx — at take 2 x4 + 4 ax® + 4 a®c + a“. 8. Take a3 – 5 a2b + 7 ab2 – 2 83 from the sum of 2 a3 9 a’b + 11 ab2 – 3 53 and 53 – 4 aba + 4 aʻb – a'. 9. Subtract a+b+c+d from e +f+g+h. 10. Take 24 – 4 xy + 6 x*y – 4 xyz + y4 from 2c4 + 4 x'y + 6 xʻya + 4 xy + y, and subtract the result from their sum. 11. Add together the given quantities in the last example, and subtract the result from 3 ** + 10 +ya + 3y*. 12. Take a’ + 12 + + 2 ab + 2 ac + 2 bc from 2 a2 + 273 + 4 ab - c. Brackets—continued. 22. It was shown, in Art. 8, that, when a quantity inclosed in brackets is to be added, we may remove the sign ( + ) of addition and the brackets without changing the sign of the terms within the brackets. On the contrary, when the quantity in brackets is to be subtracted, or has the sign minus before it, we must change the sign of every term within the brackets on removing the brackets and the sign of subtraction. We shall now see how to simplify expressions involving brackets connected by the signs of addition and subtraction ;-> Ex. 1. Simplify (3 a + 5 6) – (6 6 - 2 c) + (- 2a + b - 3c). The given expression = 3a + 56 - 66 + 20 – 2 a + b 3 c, or adding together the like quantities, = a - C. . Ex. 2. Reduce to its simplest form a - (6 – c) + {6 + (a – c)} - {(a - b) - c}. (When a pair of brackets is inclosed within another pair, it is convenient to remove the inner one first.) Hence the given expression = a - 6+e+ (6 + a-c} - {a - 6-0 3 x 2 – 7 6 x – 9 Ex. 3. Simplify the expression "4 - 2 + 8 The line separating the numerator and denominator of a fraction is a species of vinculum, since it serves to show that the whole numerator is to be divided by the whole denominator, Hence, on breaking up the two latter fractions into fractions having one term only in the numerator, we have 3 x 2 – 7 6 x – 9 3 1 7 6 9 19 4- 2 + 8 =4* – 22.+ 2 + z * - = 2 + 8 23. As it is often necessary to inclose quantities within brackets, we shall now show how this is done. The following rule needs no explanation: RULE. —When a number of terms is inclosed within brackets, if the sign placed before the brackets be +, the terms must be written down with their signs of affection unchanged; but, if the sign placed before the brackets be-, the sign of affection of every term placed within the brackets must be changed. Thus we may express a + b - C - d in any of the following ways :a + 7 - C - d = a + (6 - C - d) = (a + b) – (c + d) = A -(-6+c+ d) = (a + b – d) – c, &c. ("Then the word sign is used in future, the student is to under. staad sign of affection, unless otherwise expreşsed.) |