If 4 M. 20M.2 19,5da.) 100,25da. answer £.305 0 8,2 2 If 2 persons receive 4,625s for one day's labour, how much should 4 persons have for the work of 10,0 days? answer 4l 17s 1d. į 3 If 168 4d. be the porterage of 5,25 hundred weight for 20 miles ; what must be paid for carrying 17,75 hundred weight 7,5 miles ! answer 11 Os 8d. 4 How many men should reap 417,6 acres in 12 days, when 5 persons cut down of that quantity in half the time answer 20 men. 5 Suppose the interest of 76,941. for 9,5 months to be 15,25l. what principal will gain 61. in 12,75 months ? answer 221 11s ld. 6 When 12 oxen graze down 16,25 acres, in 20 days; how much, of like pasture, would suffice 24 such cattle for 100 days? answer 162,5 acres. 7 What money, at 3 per cento per annum, will clear 381 10s. in a year and a quarter ? answer 8801. 8 A cellar which is 22,5ft. long, 17,3 wide, and 10,25 deep, being dug in 2 days, by six men, working 12,3 hours a day; how many days, of 8,2 hours, should 9 men take to excavate one which measures 45, by 34,6 by 12,3 teet? answer 12 days. A INVOLUTION; OR THE RAISING OF POWERS. given number into itself continually a certain number of times; thus, 2x2= 4 is the second power, or square of 2. 2x2x20 8 the third power, or cube of 2. 2x2x2x216 the fourth power of 2, &c. The number denoting the power is called the index, or the exponent of that power. If two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factors; thus, 2X2-4 the square of %; 4x4=16=4th power of 2; and 16x16 256=8th power of 2, &c. N TABLE TABLE of the first nine powers, Roots Squares Cubes 4th power 17th power 8th power 9th power 11 11 11 11 1 21 4 ] 8 161 32 641 128 2561 512 31 91 271 81 243) 7291 2187] 65611 19685 4!16 641 256| 1024 40961 16384 655361 262144 5|25|125 625 31251 15625 78125 390625 1953125 61361216/1296] 77761 46656 279936] 1679616 10077696 7|49|343/2401|16807|1176491 8235431 5764801 40353607 8164 512|4096|32768|2621442097152|16777216134217728 98117296561 590491531441 4782969143046721387420489 EXAMPLES. 1 What is the fifth power of 7 ? 7x7x7x7x1=16807=fifth power. 2 What is the third power or cube of 35 ? answer 42875 3. What is the fourth power of ? 4 What is the fifth power of ,029 ? ,000000020511149 5 What is the sixth power of 5,05 ? answer 10190,005304479729 6 What is the eighth power of 34 ? 17857368 81 736 EVOLUTION: OR THE EXTRACTING OF ROOTS. CHE root of any number, or power, is such a num. , of times, will produce that power. Thus 2 is the squareroot of 4, because 2x2=4; and 4 is the cube root of 64, because 4X4X4=64, and so on. 2 THE SQUARE ROOT. * HE square of a number is the product arising from that number multiplied into itself. Extraction of the square root is the finding of such a number: as, being multiplied by itself, will produce the number proposed. RULE, 1. Distinguish the given number into periods of two figures each, beginning at the units place, or decimal point; and when the decimal does not consist of an even number of figures, annex a cypher; and equal to the periods of whole numbers and decimals respectively will be the places of each in the root. 2. Deduit from the first period the greatest square it contains, setting the root thereof as a quotient figure, and doubling it for a divisor; and bring down the second period to the remainder, for a dividual. 3. Try how often the said divisor, with the resulting fig. ure of this trial thereto annexed, are contained in the dividual, and set this resulting figure to both the divisor and root; then multiply and subtract as in division, and bring down the next period. 4. Double the ascertained root for a new divisor, and repeat the process to the end. PROOF. Square the root, adding in the remainder (if any) which will equal the number given. EXAMPLES. 30138,696025(173,605 answer 1 189 1029 3466 20969 20796 347205) 1756025 1736025 0 Notę. When one more than half the figures of the root are found, the rest inay be obtained by working as in contracted division of decimals. 2 Required the square root of 14876,2357 ? 14876,2357121,968175 22)48 44 241)476 241 2429) 23523 21861 24386) 166257 146316 24392)19941(8175 427 3 Required the square-root of 5499025 ? facit 2343 4 What is the square-root of 74770009? answer 8647 5 What is the square-root of 368863 ? 607,34092+ 6 What is the square-root of 3271,4007? 57,19+ 7 What is the square-root of 2,2710957 ? 1,50701 + 8 What is the square-root of 10? 3,162277 + 9 What is the square-root of ,0003272481,01809 10 Required the side of a square acre of land ? fasit 12,649+per. 11 A certain number of men gave 30s 1d. for a charitable purpose; each man giving as many pence as there were answer 19 men. 12 If a circular pipe of 1,5 inches diameter fill a cistern in 5 hours ; in what time would it be filled by one of 3,5 inches diameter? answer 55 min. 6 sec. men: 13 If 484 trees be planted in a square orchard, how many must be in a row each way? answer 22 Note 1. The square of the longest side of a right angled triangle is equal to the sum of the squares of the other two sides; and consequently the difference of the square of the longest, and either of the other, is the square of the remaining side. “2. The square ront of a vulgar fraction is found by reducing it to its lowest terms, and extracting the roct of the numerator, for a numerator, and of the denominalor, for a denominator. If it be a surd, reduce it to its equi. valent decimal, &c. "3. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted as before. 14 The wall of a fortress is 17 feet high, which is surrounded by a, moat 20 feet in breadth ; query the length of a ladder to reach from the outside of the moat to the top of the wall ? answer 26,2 feet. 15 A line of 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad ; the height of the wall is required ? answer 26,83+yards. 16 Suppose a ladder, 60 feet long, be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot, will reach i window 23 feet high on the other side; what breadth was the street of? answer 102,64 feet. 17 What is the square-root of 2844 ? answer 18 What is the square-root of 716? 19 What is the square-root of ,71528 20 What is the square-root of 375 61 21 What is the square-root of 1723 ? 45 22 What is the square-root of 7614 8,7649+ 6192 THE CUBE ROOT. THE Cube of a number is the product of that number multiplied into its square. Extraction of the cube root is the finding of such a number, as, being multiplied into its square, will produce the number proposed. RULE. First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal N 2 point: |