B. Expressions, which involve double brackets, reduced to their most simple forms: 1. {a-(b-c)}-{b-(c-a)}+{c−(b-a)} =a−b+c-b+(c−a)+c−(b−a) 2. ab-{(3bce-2ab) — (5bce—beƒ)+(3ab−3beƒ)} =ab-(3bce-2ab)+(5bce-bef)-(3ab-3bef) 3. 1-{1-(2-x)}+{4x-(3-6x)}+{4-(−5+6x)} Examples for Practice. Ex. IX. B. Reduce to their most simple forms the following expres sions: (1) 2a-{2a-(b+2x)}+{b− (2x− 2b)}. (2) {6a+2b-(3a+2b)} — {2a+4b−(4a—b)}. -- (3) x2-(2y3-3x2) — {2y3 — (3x2 — x2)} + {3≈2 — (2y2 — x2)}. (4) {(1+x)-(1+2x)}+{(1−x)+(1—2x)}-{(1−x)–(1—2x)}. (5) a-{b-c+x}+{b−x−2b}. (6) a2+2ab+b2 — {a2+ab-b2 - (2ub—a2 — b2)}. (7) 3x+3y2-{x2+2xy + y2 - (2xy — x2— y2)}. C. Expressions, which involve more than double brackets, reduced to their most simple forms: 1. 4x-3y-[(2x+4y)+3x+{y-9x-(2y-x)+(x−y)}] =4x-3y-[2x+4y+3x+y-9x-2y+x+x-y] 2. 11x-[7x-{8x−(9x+−6x)}] =11x-[7x-18x-(9x-6x)}]=11x-[7x-{8x-3x}] =11x-[7x-5x]=11x-2x=9x. NOTE. It is a safe method to do away with the innermost bracket first, then the next, and so on. Examples for Practice. Ex. IX. C. Reduce to their most simple forms the following expressions: (1) 2x-[(3x-9y)—{(2x-3y)-(x+5y)}]. (2) x-[x+y={x+y+x-x+y+z+w}]. (4) x-[2y+(3x-3x-(x+3)}]+{2x-(y+3x)}. D. Add together (1) a+2c-d, 3a-(b-4c)+2d, and 3a-b-(2c-d). (2) 1-{1-(1-x)}, 2x−(3—5x), and 2−(−4+5x). E. Subtract (1) ax-bx-(by—cy) from ax + bx+by+cy. (2) 3ax3-3a2x2-2ax+2a2 from 3x1+3ax3 — (2x2 — 3a2x2) — 2a2. (3) x2-y-(3c-3d2) from (3x2-3y2)-(c2-d3). (4) a-b-(2c-2d) from (2a-2b)-c+d. F. Add together (ax-bx)-(by-cy) and (ax+bx)+(by+cy), and subtract the latter from the former. 26. On the introduction of brackets or vincula into algebraical expres sions. Any number of terms of an algebraical expression may be included within a bracket or under a vinculum, thus: If the sign of the first term to be included be positive, then put the sign + before the bracket or vinculum and write the different terms now included within the bracket or under the vinculum in the same order, and with the same signs as before they were included within the bracket or under the vinculum. If the sign of the first term to be included be negative, then put the sign- before the bracket or vinculum, and write the different terms now included within the bracket or under the vinculum in the same order as before, but with all their signs changed from + to —, and from to +; because the sign before a bracket or vinculum signifies that all terms within the bracket or under the vinculum are to be subtracted, i.e. to have their signs changed when the bracket or vinculum is removed. Thus a+b-c+d=a+(b-c+d), or =a+b-(c-d). a-b+c-d=a-(b−c+d), ora-b+(c-d), or =a−b+c−(d). a+b-c-d=a+b−(c+d), =a+{b-(c+d)}. a-b+c+d=a-b+(c+d), ora-{b-(c+d)}. Ex. X. Express a +26-3c+4d-5e+6f-7g+8h in brackets: 1. Taking the terms two together. 2. Taking the terms four together. 3. Including the last two terms in each bracket of the last result in an inner bracket. 4. Including all the terms after the first two in a bracket. 5. Resolving the last five terms of the last result into two brackets, the first of two terms, the second of three terms, so that every bracket in the expression may be included in the one which goes before it. MULTIPLICATION. 27. First, of Simple Algebraical Quantities. RULE. "Multiply the numerical coefficients of the two quantities together, and write the letters in order after the product of the numbers, prefixing to the product the sign + if the multiplicand and multiplier have the same sign, and the sign - if they have different signs." Ex. 1. 5a × 2c=10ac. Ex. 2. 15x2x-3y2= -45x3y3. Ex. 3.3mn x-1pq=mnpq. Secondly, of Compound Algebraical Quantities. RULE. "Multiply every term of the multiplicand by each term of the multiplier, by the last Rule, beginning at the left hand and proceeding to the right; then connect the several products together by the rules of Addition, and that sum will be the product required." Obs. 1. When factors are multiplied together the product is the same in whatever order the operation is performed: thus 4×6 is the same as 6 × 4, for 4×6 may represent 4 objects taken 6 times or 24 objects, and 6 × 4 may represent 6 objects taken 4 times or 24 objects. See Arith. Art. (32). X On the same principle ab is equivalent to ba, and abc, acb, bca, cba, bac, cab, are all equivalent. Obs. 2. When factors contain different powers of the same quantity, it is advantageous, for the purpose of making like quantities in the several partial products fall as much as possible in the same columns, to arrange the terms according to the indices of those powers. 28. It has been stated that "If the multiplier and multiplicand have the same sign, the product is positive; if they have different signs, the product is negative." The rule of signs in the product of (a−b) and (c-d), if a>b and c>d, may be thus deduced. To multiply (a−b) by (c-d) is to add a-b to itself as often as there are units in c- -d; therefore it is the same as (a−b) taken c times diminished by (a-b) taken d times. Now (a-b) taken c times is the same as a taken c times diminished by b taken c times, or ac diminished by bc, and ac-bc. Similarly (a-b) taken d times = ad- bd, therefore (a−b) (c-d)=ac-bc-(ad—bd) =ac-bc-ad+bd by rule of Subtraction. If a<b and c<d, the rule of signs in the product is assumed to be true. 29. The product of two or more powers of the same quantity is expressed by writing that quantity with an index equal to the sum of the indices of the proposed powers. The reason of this is evident when the powers are positive integers. For a2 is the same as a × α, a3 is the same as a×a×a; therefore a3 xa2= (a×a×a) × (a×a) =αχαχαχαχα =a (by def.) Similarly a×a" a×a...to m factors xaxa...to n factors = a×a.....to (m+n) factors Obs. We have shewn that am × a" = am+n when m and n are positive integers: when m and n are either or both negative or fractional, the above Rule is assumed to hold good. A. Examples in multiplication, where the multiplicand is a compound quantity, and the multiplier a simple quantity, worked out. |