17. —[m−(m+n) — (m − n) − (− m+n)]. 19. 2x-{x-(xy)-[x-x-y]-y}. 20. 12-2a-{-a-[2a-(a-7 — a)]}. 21. 14-3a-9 a-[10 a -(11a - 6 — 6 a)]}. 22. a-[-{-(— a)}]. 23. a-3-[-{− (− a + a + b)}] 24. x+y-[-(x − y) + {− x+(x − x − y)}]. 25. 1—{—a—(a + 1) − [− a − (a — a − 1)]}. 26. a-(-{-[—(− a)]}). 27. 1-(-{a+(-a + 1)}) — {a — a −1}. 28. 6m+{4m− [8 n − (2 m + 4 n) — 22 n] — 7 n } +[9m-(3n+4 m)+14n]. 29. 1247-[1722 - {1722 +(933 — 1247)} ]. 30. From a +{(4 − b ) + ( a − 4) — a −7} subtract a — { (6 − b) + (6 a − 6) – (5 a — 7)}. 31. From the sum of a +{a − (b −c)} and - a +[4 a−(5b+c)] subtract a − (b — c). 32. Simplify 4 a − [6 b + (3 a −c) — {5 b— c — a}] and check the answer by substituting a=3, b=2, c=1 in the question and the answer. 33. Simplify 9a-[-7a+5b — (a−b) + a−b}] and check the answer by the substitution a = 1, b = 2. 46. Signs of aggregation may be inserted according to § 43. Ex. 1. In the following expression inclose the second and third and the fourth and fifth terms respectively in parentheses: a−b+c+ 2 d· e =a (b−c)+(2 d − e). Ex. 2. Inclose in a parenthesis preceded by the sign the last three terms of 2a+b-5c+2 d EXERCISE 16 In each of the following expressions inclose the last three terms in a parenthesis: 1. e+b+c-d. 2. x-2y+3z-4 d. 3. 2x-6x2-7x3+5x1. 4. 2p-q+p-q2. In each of the following expressions inclose the last three terms in a parenthesis preceded by the minus sign: Write the following expressions: 1. The sum of the squares of a and b. 2. The square of the sum of a and b. 3. The difference of the cubes of x and y. NOTE. The minuend is always the first, and the subtrahend the second, of the two numbers mentioned. D 4. The difference of the cubes of y and x. 5. The cube of the difference of x and y. 6. The product of a and b. 7. The product of the cubes of a and b. 8. The cube of the product of a and b. 9. The product of the sum and the difference of a and b. 10. Six times the square of the difference of a and b diminished by the quantity a minus b. 11. The product of the difference of a and b and the quantity a22b2+ c3. 12. The sum of a and b diminished by the difference of a and b. 13. a cube minus the quantity 2 x2 minus 6 y2 plus 7 c2 plus the quantity -x+y. 14. The sum of the cubes of a, b, and c divided by the difference of a and c. Write algebraically the following statements: 15. The sum of a and b multiplied by the difference of a and b is equal to the difference of a2 and b2. 16. The difference of the cubes of a and b divided by the difference of a and b is equal to the square of a plus the product of a and b, plus the square of b. 17. The difference of the squares of two numbers divided by the difference of the numbers is equal to the sum of the two numbers. (Let a and b represent the numbers.) CHAPTER III MULTIPLICATION MULTIPLICATION OF ALGEBRAIC NUMBERS EXERCISE 18 1. If a man makes $15 a day, how many dollars will he make in 5 days? 2. If a man loses $15 a day, how many dollars will he make in terms of algebra in 5 days? (Denote gain by +, and loss by -.) 3. If from a man's fortune $15 are deducted 5 times, how much in terms of algebra does he make? 4. If from a man's debts $15 are deducted 5 times, how much does he gain? 5. Express each of the Exs. 1-4 as a multiplication example, considering gain as positive, and loss or debts as negative. 6. If we denote three days hence by +3, by what must we denote three days ago? 7. If we denote northerly motion as positive, and three days hence as +3, express the following as multiplication examples with algebraic symbols: (a) A ship sailing north at the rate of 3° per day and crossing the equator to-day will, in 6 days, be 18° north of the equator. (b) The same ship 6 days ago was 18° south of the equator. (c) A ship sailing south at the rate of 3° per day and crossing the equator to-day will, in 6 days, be 18° south of the equator. (d) The same ship 6 days ago was 18° north of the equator. 8. If the signs obtained for the products in the preceding examples were generally correct, what would be the value of 6 x 3, 6 (-3), (−6) × 3, (−6) × (−3)? 9. State a rule by which the sign of the product of two numbers can be obtained. 47. Multiplication by a positive integer is a repeated addition; thus, 4 multiplied by 3, or 4 x 3=4+4+4=12, - 4 multiplied by 3, or (−4) × 3 =(−4)+(−4)+(−4) = −12. The preceding definition, however, becomes meaningless if the multiplier is a fractional or negative number. To take a number 2 or -7 times is just as meaningless as to go to bed 2 times or to fire a gun -7 times. A more useful definition of multiplication which may be used in nearly all multiplication problems, is the following: 48. Multiplication is the operation of finding a number that has the same relation to one factor (multiplicand) as the other factor (multiplier) has to 1. Thus is obtained from 1, by taking one fifth of unity three times, or 용=1++. Therefore 6 multiplied by & is obtained by taking one fifth of 6 three times, or 49. The product of 4 multiplied by 3 is obtained from 4 in the same manner in which · 3 is obtained from 1. But -3=-1-1-1, i.e. -3 is obtained by subtracting three times 1. Therefore, (+4) × (− 3) = −(+4) −(+4) −(+4)= − 12. (−4) × (− 3) = −(− 4)—(−— 4)—(− 4) =+ 12. and |