of 2, &. The Double Rule of Three in Decimals. 145 H 3M. 20M.2 answer 6.305 o 8,2 19,5 da 2 If 2 persons receive 4,625s. for day's labour, how much should 4 persons have for the work of 10,5 days? answer 41 175 id.is 3. If 16s 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 il 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 zo men, 5 Suppose the interest of 76,941. for 9,5 months to be 15,257. what principal will gain 6l. in 12,75 months ? answer 22l 11s ideas 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 31 per cent. per annum, wili clear 381 105. 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 2ļ days, by six men, working 12,3 hours a day; how many days, of 8,2 hours, should omen take to excavate one which measures 45, by 34,6 by 12,3 feet? answer 12 days. INVOLUTION; OR THE RAISING OF POWERS. POWER is the product arising from multiplying any of times ; thus, 2X2= 4 is the second power, or square of 2. 2X2 X 2= 8 the third power, or cube of 2x2x2x2=16 the fourth power The number denoting the power is called the index, or the exponent of that power. If two or more powers are multiplied together, their pro. du& is that power whose index is the sum of the exponents of the factors; thus, 2X2=4 the square of 2; 4X4=16=4th power of 2; and 16x16=256=8th power of 2, kc. N TABLE A on 1 1 1 1 11 11 11 11 1 1 21 41 81 161 321 64 1281 2561 512 31 9 271 811 243 7291 2187! 65611 19683 4116 641 2561 1024 4096 16384/ 655361 262144 5|25|125 6251 31251 156251 78125 390625 1953125 613621612961 777646656| 279936| 1679616 10077696 7149;34312401/16807/117649 823543 5764801 40353607 8|645124096|32768|26214412097152|16777216134217728 981729|6561 59049531441/4782969|43046721|387420489 E X A M P L E S. 91 | What is the fifth power of 7 ? 7x7x7x7x7=16807=fifth power. 2 What is the third power or cube of 35? answer 42875 3 What is the fourth power of 1? TIT What is the fifth power of ,000000020511149. Ś What is the sixth power of 5,03? answer song 005304479729. 6 What is the eighth power of 3? 1785700615 4 2029? 366816 EVOLUTION: OR THE EXTRACTING OF ROOTS. TH HE root of any number, or power, is such a numi ber, as, being multiplied into itself a certain number of times, will produce that power. Thus 2 is the square root of 4, because 2 X 2=4; and 4 is the cube root of 642 because 4X4X4=64, and so on. THE SQUARE ROOT. HE square of a number is the product arising from TH 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 num. ber 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. Deduct 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 figure 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. E X A M P L E S. 1 What is the square root of 30138,696025 ? 30138,696025(173,605 answer. . 1 189 1029 3466120969 20796 367205)1736025 1736025 Note. When one more than half the figures of the root are found, the reft may be obtained by working as in contracted division of decimals. 1 Required the square-foot of 14876,2357? 3 Required the square root of 5499025? facit 2345 4 What is the square-root of 74770609? answer 8647 5 What is the square-root of 368863? 607,34092+ Ó What is the square-root of 3271,4007? 57,19+ What is the square-root of 2,2710957? 1,50700+ s What is the square-root of 10? 3,162277+ 9 What is the square-root of ,0003272481? ,01809 jo Required the side of a square acre of land? facit 12,649 +per. It A certain number of men gave 30s id. for a charit. able purpose ; each man giving as many pence as there were men ; query the number? answer 12 If a circular pipe of 1,5 inches diameter fill a cis. tern in 5 hours ; in what time would it be filled by one of 3,5 inches diameter? answer 55min. 6sec. 19 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 fum 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 fide. 2. The square root of, a vulgar fraction is found by reducing it to its lowest terms, and extrading the root of the numeracor, for a nume. rator, and of the denominator, for a denoninacor. If it be a surd, reá duce it to its equivalent 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 a window 23 feet high on the other side ; what breadth was the street of ? answer 103,64 feet. 17 What is the square-root of 2:44 18 What is the square-root of 2.15? 39. What is the square root of its 271528 20 What is the square root of 3745? 67 21 What is the square-root of 171? 44 22 What is the square-root of 76? 8,7649+ 2 THE THE CUBE ROOT. multiplied into its square. Extraction of the cube root is the finding of such a num. ber, as, being multiplied into its square, will produce the pumber proposed. RULE. First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal N 2 point; |