round, (2) completely round the garden, supposing the pathway occupies one eighth of the space? 4. Two persons start from the place where two roads meet at right angles to each other, one walks 3 miles an hour and the other 4; how far distant are they from each other 2 hours and 45 minutes after they started? 5. Find the length of a footpath which crosses diagonally a rectangular field 143 yards long and 116 broad. 7. x-6a+15a-20a3+15a2- 6a+1. 8. 27a-108ax+171ax2-136a3x+57a2x1-12ax3+x®. 9. 8x-36x+66x-63x2+33x2-9x+1. 10. XI. Find the cube roots of the following numbers 8 23+12-623. 1. 262144, 531441, 1953125, 3048625, 4492125, 5177717. 2. 15625, 140608, 1677216, 277167808, 16915218263, 448048351808. 3. 064, 000064, 009261, 000405224, 000830584, 000027270901. 4. 1·01, 9.6, ·4, 04, 004, 21·1, 3·43, 8.88, 9.04, 1·912, 10-001. 5. 1.030301, 884-736, 9393-931, 40-353607, 700-227072, 738763-264 6. 2, 7, 10, 100, 3.14159, each to three places of decimals. 7. 1, 7, 4, 33, 3, 1000, 219, 3. XII. 1. The second differences of the cubes of the natural numbers form a series of numbers whose differences are a constant number. What is that number? 2. The cube root of every number greater than 3 is greater than the fourth root of the number increased by unity. 3. Prove that no number, the sum of whose digits is 6, can be either a perfect square, a perfect cube, or the difference between two perfect cubes. 4. The cube of every odd number greater than unity can be expressed in two different ways by the difference between two numbers which are perfect squares. 5. Shew that the cube root of an exact cube containing four, five or six figures can be determined by inspection, by taking the cube root of the first figure, or of the first two or three figures, and observing with what two figures the number ends. 6. Show that the cube root of a number not a complete cube cannot be represented by any rational fraction, and if it be reduced to a decimal the latter cannot have a recurring period. 7. When will the cube root of a recurring decimal be a recurring decimal? Give examples. 8. Find the fourth root of 00028561, the sixth root of 000004826809, and the ninth root of ⚫000000010604499373. XIII. 1. Every integral number consisting of n digits has 3n, 3n-1, or 3n-2 digits in its cube. 2. The difference of the cube of the sum of any three numbers, and the sum of the cubes of the numbers, is always exactly divisible by the sum of any two of the numbers. 3. If two numbers differ by a unit, their product, together with the sum of their squares, is equal to the difference of the cubes of the two numbers. 4. Show that any cube number when divided by 4 or by 7 cannot leave 2 for a remainder. 5. Shew that the product of any three consecutive numbers increased by the second number is always a perfect cube number. 6. The sum of the cubes of two integral numbers increased by unity is greater than three times their product. 7. Decompose 15750 into its prime factors, and find the number into which it must be multiplied to make it a perfect cube. 8. Determine an algebraical expression by which a series of cube numbers can be found, each of which shall be the sum of three cube numbers. XIV. 1. If the edge of a cube be four lineal inches, what is the number of square inches in its surface, and the number of cubic inches in its volume ? 2. If the volume of a cube contain 1,000 cubic inches, find its surface, the length of one of its edges, the diagonal of one of its faces, and the diagonal of the cube itself. 3. If the edge of a cube be 12.75 inches, what is the length of the edge of another cube which is twice the magnitude of the first? 4. The diagonal of a cube is one foot longer than each of the edges; what is the content of the cube in inches? 5. What is the volume of a cube whose diagonal is 12 lineal inches? and what is the diagonal of a cube whose volume is 12 cubic inches? 6. If the surface of a cube be 50 square inches, what is its volume? and if the volume be 50 cubic inches, what is the surface? 7. The three conterminous edges of a rectangular parallelopiped are 36, 75, and 80 inches respectively; find the side of a cube which shall be of the same capacity. 8. What is the content of that cube, the surface of which is 900 square feet 54 inches? 9. How many cubes having the area of one side 24 square inches, are equivalent to that cube, an edge of which is 36 inches? 10. Having given a cube whose side is 100 inches; find (within the tenth of an inch) the side of a cube whose magnitude is double of the former cube. Why cannot the problem be solved exactly? 11. Find the length of the edge of a cube whose volume shall be equal to three cubes whose surfaces are respectively 2,400, 5,400, and 15,000 square inches. 12. If the lengths of the edges of three cubes be a, b, a+b; shew that (a+b)3 2 XV. 1. What must be the dimensions of a chest whose length is double its breath, and the breadth double its depth, that its content may be a million cubic inches? 2. The content of a cube is 1,012,199,273,930,125 cubic feet; how many square yards of canvas would be required to cover it? 3. The internal surface of the bottom of a box one foot deep, contains eight square feet; find the edge of a cubical box of the same content. 4. The breadth of a room is twice its height, and half its length; and the content is 4,096 cubic feet; find the dimensions of the room. 5. The area of floor of a room is 768 square feet, that of each of the two opposite walls 576 square feet, and that of each of the other two walls 432 feet; find the height and content of the room. 6. What must be the length of the side of a square cistern four feet deep, in order that it may contain 5,000 gallons of water, supposing a gallon to contain 272 274 cubic inches? 7. If a cubic foot of water weigh 1,000 ounces; find the dimensions of a cubical tank what shall contain a ton of water. 8. If the external edge of a cubical chest be a inches, and the chest be composed of boards m inches thick; find the difference between the external and internal surfaces of the chest, and the content of the boards of which it is composed. 9. Find the length of the edge of a cubical block of coal which weighs 17 tons 12 cwt., if a cubic foot of coal weigh 77 pounds. 10. The cost of a cubic mass of metal is £8,584 18s. 7d. at £4 3s. 4d. a cubic inch; what is the area of one face of the mass? 11. A hollow cubical box made of material 1 inches thick has an interior capacity of 50-653 cubic feet; determine the length of the outside edge of the box. 12. A cubical box a inches deep is filled with layers of spherical balls, each one inch in diameter; find the number contained in the box, and what portion of space in the box is vacant. Find also what portions of the space would be vacant, (1) when each of the balls is half an inch, and (2) when one-third of an inch in diameter. XVI. Reduce the following expressions to their simplest forms:1. am+" xam-". 2. aTM+÷am―n 3. am-"÷am+n ̧ 4. a"-"xa"-P × a2-m. 5. am- Xan-r÷a”¬¶ ̧ 6. a2m × a-3m × a”. 7. a3÷a2-". 8. a-3a"-" 9. (a)(a)−3m. 2m-n 22. {{aTM) ̄*}" × {(a−2m)3"}−2. 23. {a.q}===× (a"+"a")". {a.a ̄) 21. _{(ab)"}=" (a=b-m)m 24. (a ̄"a"+")='÷(a ̄"a"-x)=. XVII. Verify the correctness of the following equivalents : = a5 bi 5 1. {—a". (—b)'.(—c)~°}3 — { a***}". 2. {x2(—y°)3÷—y3 ( − x2)$}6 — — x3y3. 3. {a'b3c(a2b3c)}} = ab1ct. 4. {(ab3)1. (a3b3)*. (a3b1)* } is = ar«b*. 6. {(ab2)3·(a2b)3. (a3b)* . (a2b1o)3 } = (a2, b1)t. {}{}{ = 3° . a . ( a + b) " . ( a + x) * } * = 9 (a+b) = { 27.ax-3 "a" (a + x) ** 6. (a3-bys. 7. (a3+b3)(a3—a3b3+b3). 8. (a*b*)(ax+a*b*+b3). 9. (a*+a1b3+b})(a} — a3b}+b}). 10. (x*-2x+3)(x*+2x+3). 11. (a}+a2b1+a}b}+ab+a+b}+b})(aa—b*). 12. (x+2y+3)(x-2y++32*). 14. (x-3x+6-3x+x-3)2. XIX. 13. (a}+3a1b§+9ba)(a3 — 3b3). Determine the following quotients, and verify the truth of them:1. a-b by a+b. 2. a3-b3 by a1-b. 3. a2+b2 by a+21a1b1+b. 4. a1+b1 by a}+b§. 5. a§—b§ by a1+b1. 6. 3ao—4alb}+bi by (al—bi)3. 7. x-3x+9x+-3 by x-3. 8. xl+yl-z-2xlył by x-yi+zl. 9. x}—x2y1+x}y—xy +x1y2—y} by x2+xy+y2. 10. a2-ab+b2 by a-31ab+b. 11. x-4x+12x-9 by x-2x+3. 12. +1 by x2-2x+1. 13. 2x3-6x+5 by 2x+21+1. 14. x'y' by x1—y1. Divide -y by x-y, and x-y by x-y, and from the results infer the quotient of 25-y5 divided by x-ył. XX. Verify the following equivalent expressions: 1. (x-2x+1)(x+21x1+1) = x2+1. 2. (a2+21ax+x2)(a3 − 21ax+x2) = a*+x1. 3. (x2+3x+1)(x2-31x+1)(x−21x1+1)(x+211+1)=x+1. 4. (x2+21x+1)(x2 — 2*x+1)(x2 —21x—1)(x2+2*x—1) = x2-4x+2x-4x2+1. 5. (x−1){x+(1−√3)}.{x+(1 − √/3) } = x3 +x2-4x+2. 6. {x2+1(1+√5)x+1}.{x2+1(1−√5)+1} = x*+3x2+1. 7. (x+√2−√3)(x−√/2−√3)(x+√2+√3)(x−√2+√3) |