4. Separate a8 — 38 into prime factors. 5. A and B can do a piece of work in a days, A and C in b days, B and C in c days. In how many days could each person do it ? 6. What is the rule for multiplying together different powers of the same letter ? For dividing ? Explain the season. Multiply a5 by aʼ; am by a". Divide as by a; aby ab; a3 by ao; am by a"; 6 a by 2 a. 7. Divide 25 - 45 by u — y. yo x 8. Find the seventh power of 3 a-- 26 by the Binomial Theorem. XVII. 1. Reduce the following expression to its simplest form: (22 + y)2 — (x + y) ( [z — y] - [z — x]). []y 2. What is the reason that, when different powers of the same quantity are multiplied together, the exponents are added ? cm+n x acman = what ? cum +n man = what? Give the square root of each of these results. 3. Resolve the following expression into a single fraction (finding the least common denominator, and reducing the 4 2cm y answer to its lowest terms): 1. What 22 — 2 + y 100 a4 72 5 ao b . 378 is the most reduced value of 5 a 6 + 368 - 1 ? 25 at 12 976 a4 72 24 a® x 78 32 x2 ya 4. Divide by 5. The owners of a certain mill make a dollars a day each, sharing equally. If the number of owners were 6 less, they would make c dollars each. Required the number of owners and the total daily profit of the mill. are the answers if a= 80, 6 3,c= 50 ? 6. Solve the equations 37 + 3x - 12 y = 8z + 55, 9 jy X = 42. 9 2 2 7. Solve the equation 1 3r 1= 8. Find - b (a — ) and (* - y)" by the Binomial Theo rem. XVIII. 1. Reduce the following expression to its simplest form: (a + b) a - ((a - b)2 – (b a) b). - – 6 — 2. Separate no n into its prime factors. 1 + x2 1 x2 3. From subtract and divide the result by 1 1+ 37 4x 1 + x2 4. “In multiplication and division, like signs give plus and unlike signs give minus.” Explain fully why this x2 is so. 5. A can perform a piece of work in a days, B can perform the same in b days, and C in c days. In how many days will the work be performed if they all labor together ? 6. Solve the equations y tä y 2 3 + 5, 3 5 4 5 2y - 5 C X 2 + 3 X 10 = 2. 3 2 5 8. Find (a +5)* and (1 — s)" by the Binomial Theo b5 1 208 3 rem. XIX. . a x2 + xy, and reduce the X 1. Reduce the following expression to its simplest form: (9 a2 32 — 464) (a? — 52) — (3 ab — 262) (3 a [a2 + 62] — — 26 [32 + 3 ab - a]) b. 2. Divide 36 za +1-64x4 - 12 x by 6x-1-82 – – 2c. 3. What is the reason that when different powers of the same quantity are multiplied together their exponents are added ? 4. Reduce to one fraction with the lowest possible denom 3a + 26 25 a? 72 inator a +6 a2 72 26 2 + y 5. Divide #* - 2xy + y by y answer to its lowest terms. 6. Find win terms of a, b, and c, from the equation bc What is the value of w when a = 2, b b=-1,c= 3 ? 7. A man bought a watch, a chain, and a locket for $216. The watch and locket together cost three times as much as the chain, and the chain and locket together cost half as much as the watch. What was the price of each? 8 - 3 x 8. Solve the equation * + 12 1 b a - 2 x CX a 5 oC x $-$-1 . 9. Find (a — )and (xy-x)" by the Binomial Theo rem. XX. 8 ao ax a a X 1. Separate into prime factors 206 — X. 2. Reduce to its simplest form 3 a5 — 4a8 + 26 – ca 3 (a? — 1) + {26 — [7a6 — a3 (4 c) — 25 (4 + c)]}. - ab , 3. Divide x2 + by x, and subtract the a2 x2 a? quotient from 4. It is said that when a term is transposed from one member of an equation to the other, its sign should be changed. Why is this so ? 5. A reservoir is supplied by two pumps. Both pumps were worked three hours and the reservoir was found to be half full. On another occasion the larger pump was worked two hours and the smaller seven hours, when the reservoir was found to be two thirds full. hours required by either pump alone to fill the reservoir ? 6. A laborer, having built 105 rods of stone fence, found that if he had built two rods less a day he would have been six days longer in completing the job. How many rods a day did he build ? 7. What is Elimination ? Describe fully the several processes by which it can be effected, and illustrate by examples of your own selection. 8. What is the Binomial Theorem? Find the seventh power of ja- 4bc by aid of it. How many XXI. 1. Reduce to its simplest form the following expression: (a + b) x — (6-0)-[(b - x) 6 - (- 0) (b + c)] xb a X. 2 c 25 a2 18 x2 10 a 78 2. Divide by 18 ya 27 e x y8 3. Divide 8 at — 22 a8b + 43 a 62 — 38 a b3 + 2484 by 2 a_ 3 ab + 46%. 4. Separate a8 — 208 into its prime factors. 5. Reduce to its simplest form the following expression : 6. Find, by the Binomial Theorem, the sixth term in the development of (a — 6)18; and the fourth term in the de 3 227 velopment of (2 2 2 ? 6% = y + 4 3 + 7. Find the values of x, y, and %, from the equations 3 y 1 42 5 3 x + 1 6 7 + } 14 6 21 8. A person performs a journey of 192 miles in a certain number of days; had he travelled 8 miles more a day he would have performed the journey in two days less time. Find how many days it took him to perform the journey. 9. Solve the equation (x - 1) (x - 2) = 6, and verify the results. XXII. 1. Reduce to its simplest form the expression a - (26 + [3c — 3a — (a + b)] + 2a - (b + 3c)). 2. Separate into its prime factors the expression 28 — yo. 3. Divide (a— bc)3 + 863 c3 by a2 + bc. 4. Solve the equation (a + x)(8 + x)=(c + x) (d + x). 5. A can build a wall in one half the time that B can; B can build it in two thirds of the time that can; all to |