EXAMPLE. To extract the square root of 346. Reducing to an equivalent decimal, we find .125. We have then to extract the square root of 346.125. 3'4 6.1 2'50 (18.6 044 1 28)246 224 366) 2212 2196 37204) 165000 148816 372084) 16 18 400 1 48 8 3 36 130064 We separate the integer 346 into periods from right to left, and the decimal .125 into periods from the decimal point towards the right, and annex a 0 to complete the last period 50. All the periods in the given number having been included in the operation, we annex 00 to the remainder, and thus continue the operation. The two decimal periods .12′50, with the two periods of Os annexed to the remainders, make four decimal periods; hence we make four decimal figures in the root. The product of two decimal fractions contains just as many decimal figures as are in both the factors. The square of a decimal fraction has, therefore, twice as many decimal figures as the decimal itself. Hence, each decimal period must contain two figures; and the number of decimal figures in the root must equal the number of decimal periods. $302. In extracting the square root of an approximate decimal, such decimal should be continued to twice as many figures as the number of decimal figures required in the root. For example, to extract the square root of to three decimal figures. As is an imperfect square, we reduce it to a decimal of six figures. =.333333 The approximate decimal thus found, having three periods, will give three decimal figures in its square root. EXERCISES. Ans. .763. Ans. .674. Ans. .892. Ans. .5897'. Ans. 27.16'. Ans. 6.613'. Ans. .0857'. 11. Find the square root of .582169. 12. Find the square root of .454276. 13. Find the square root of .795664. 14. Find the square root of .3478312. 15. Find the square root of 737.8742. 16. Find the square root of 43.73731. 17. Find the square root of .0073474. 18. Find the square root of 2318. 19. Find the square root of 18. 20. Find the square root of 23734,28% 21. Find the square root of 74786 346 22. Find the square root of 90374376. 23. Find the square root of 23473783. 24. Find the square root of 8473763. Ans. 920.530'. 25. Find the square root of 78370317. APPLICATION 4000. 7348 § 303. The area of a surface is equal to the product of its length into its breadth; using these dimensions in the same denomination. Since a square has its length and breadth equal to each other, the area of a square is equal to either of its sides multiplied into itself; that is, the area of a square is the square of either of its sides. Hence, the area of the square being given, a side of the square will be found by extracting the square root of the area. EXERCISES. 26. How long must the side of a square lot be, which shall contain just one acre of ground? Ans. 12.649' p. 27. How long must the side of a square field be, which shall contain just 10 acres of ground? Ans. 40 poles. 28. What must be the side of a square, which shall be equal in area to a surface 320 yd. long, and 75 yd. wide? Ans. 154.919' yd. 29. A merchant bought a bale of cloth, containing just as many pieces as there were yards in each piece. The whole number of yards was 1089; what was the number of pieces? Ans. 33 pieces. 30. What must be the sides of two squares, one of which shall contain 2 square miles, and the other 3 square miles, of land? Ans. 452.548' p.; and 554.256' p. 31. A regiment consisting of 5476 men, is to be formed into a solid square. How many men must be placed in rank and file, that is, in a line on the front, and from front to rear? Ans. 74 men. 32. What would be the expense of enclosing 15 A. 2 R. 18 P. of ground, in the form of a square, at the rate of $2.12 per rod for the fencing? Ans. $424.82'. 33. A company of men on a journey expended $6084; each man expending as many dollars as there were men in the company. What was the number of men? Ans. 78 men. 34. A farmer wishes to plant an orchard which shall contain 8464 trees, and have as many rows of trees as trees in each row. What will be the number of trees in each row ? Ans. 92 trees. 35. A city whose corporate area is in the form of a circle, contains 3.1416 square miles, and is 6.2832 miles in circumference. Had the same amount of area been incorporated in the form of a square, what would have been the compass of the city? Ans. 7.088' miles. 36. What must be the dimensions of a field to contain 10 acres of ground, and have its length equal to twice its breadth? One half of the given area will be the area of a square, whose size is equal to the breadth of the field. Ans. 28.284' p. in breadth; 56.568' p. in length. 37. A garden which shall contain an acre of ground, is to have its breadth equal to one half of its length. What must be its dimensions? Ans. 8.944' p. in breadth; 17.888' p. in length. 38. A cemetery containing 15 acres is laid out in such a manner that its length is equal to three times its breadth. What are the dimensions of the cemetery? Ans. 28.284' p. in breadth; 84.852' p. length. 39. A warehouse whose base shall occupy 10000 square feet, is intended to have its breadth only one-third of its length. What must be the length and breadth? Ans. 173.205' ft. by 57.735' ft. 40. A farmer intending to enclose 50 acres of land, wishes to know what difference in the amount of fencing there would be between having the enclosure in the form of a square, and having it such that its length shall be double its breadth? Ans. 21.702' poles. Extraction of the Cube Root. The following principles form the basis of the Rule for extracting the cube root. § 304. I. The cube of a number has, at most, only three times as many, and, at least, only two less than three times as many figures, as the number itself. Thus, 993-970299; 9993-997002999; in which examples, the cubes have only three times as many figures as the respective numbers or roots, and those numbers are the largest that can be expressed by the same number of figures. Again, 103-1000; 1003-1000000; in which the cubes have only two less than three times as many figures as the numbers or roots, and those numbers are the smallest that can be expressed by the same number of figures. From the preceding, it follows that § 305. II. A number has three figures for each figure in its cube root, excepting the left hand one-for which it has one, two, or three figures, according as said left hand figure produces one, two, or three figures, in cubing the root. § 306. III. If a number be divided into any two parts, the cube of the number will be equal to the cube of the 1st part, + 3 times the square of the 1st into the 2d, + 3 times the first into the 2d2, the cube of the 2d part. For example, 16=10+6, and the cube of 16 = the cube of (10+6). The square of 16, equal to the square of (10+6) as heretofore found, is 100+ 120+ 36 600+ 720+216 1000+1200+ 360 1000+1800+1080+216 the cube of (10+6)=163. The square of (10+6) multiplied by (10+6), produces the cube of (10+6). Multiplying, we have 36×6=216; 120×6=720; and 100 ×6=600; also, 36×10=360; 120×10=1200; and 100X10 =1000. The sum of all these products is equal to 1000+1800+1080 +216=163. This cube is composed of 103=1000; 3 times 102×6=1800; 3 times 10X62=1080; and 63=216. RULE LVIII. § 307. To extract the CUBE ROOT of a given number. 1. Separate the given number into periods of three figures each, from right to left; observing that the last period may sometimes have but one or two figures. 2. From the left hand period subtract the greatest cube number it contains, and set the root of said cube for the first figure of the root required. 3. To the remainder affix the next period for a dividend. Divide this dividend, exclusive of its two right hand figures, by three times the square of the root already found, taken as an incomplete divisor, and annex the quotient figure to the root. 4. Complete the divisor by annexing to it two Os, and adding three times the product of the last quotient figure into the previous root with a 0 annexed, and also the square of the last quotient figure. 5. Multiply the divisor thus completed by the last quotient figure; subtract the product from the dividend; to the remainder affix the next period for a dividend; divide by three times the square of the root already found, and so on, as before, until the operation is completed. In applying this Rule it will be convenient to refer to the following The left hand period is 91, and the greatest cube number it contains is 64, the root of which is 4. Subtracting, and to |