CHAPTER XIII. POWERS AND ROOTS. 263. The square of a number is the product of two factors, each equal to this number. are Thus the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 264. The square root of a number is one of the two equal factors of the number. are Thus the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 265. The square root of a number is indicated by the radical sign √, or by the fraction written above and to the right of the number. 266. Since 35=30+5, the square of 35 may be obtained as follows: 267. Hence, since every number consisting of two or more figures may be regarded as composed of tens and units, The square of a number will contain the square of the tens + twice the tens × the units + the square of the units. = SQUARE ROOT. 268. The first step in extracting the square root of a number is to mark off the figures of the number in groups. Since 112, 100 102, 10,000 = 100', and so on, it is evident that the square root of any number between 1 and 100 lies between 1 and 10; of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any integral number expressed by one or two figures is a number of one figure; expressed by three or four figures is a number of two figures, and so on. If, therefore, an integral number be divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number of groups of figures. The last group to the left may have one or two figures. Find the square root of 1225. 12 25 (35 9 65)3 25 3 25 The first group 12, contains the square of the tens' number of the root. The greatest square in 12 is 9, and the square root of 9 is 3. Hence 3 is the tens' figure of the root. The square of the tens is subtracted, and the remainder contains twice the tens X the units + the square of the units. Twice the 3 tens is 6 tens, and 6 tens is contained in the 32 tens of the remainder 5 times. Hence 5 is the units' figure of the root. Since twice the tens the units + the square of the units is equal to (twice the tens + the units) × the units, the 5 units are annexed to the 6 tens, and the result, 65, is multiplied by 5. 269. The same method will apply to numbers of more than two groups of figures, by considering the part of the root already found as so many tens with respect to the next figure. root of 7890481. Extract the square 7 89 04 81 (2809 4 48) 3 89 384 5609) 5 04 81 5 04 81 When the third group, 04, is brought down, and the divisor, 56, formed, the next figure of the root is 0, because 56 is not contained in 50. Therefore, O is placed both in the root and the divisor, and the next two figures, 81, are brought down. 270. If the square root of a number have decimal places, the number itself will have twice as many. Thus, if 0.11 be the square root of some number, the number will be (0.11)2 = 0.11 × 0.11 0.0121. Hence, if a given number contain a decimal, we divide it into groups of two figures each, by beginning at the decimal point and marking toward the left for the integral number, and toward the right for the decimal. We must be careful to have the last group on the right of the decimal point contain two figures, annexing a cipher when necessary. Extract the square root of 52.2729. 52.27 29 (7.23 49 142) 3 27 284 1443) 43 29 43 29 It will be seen from the groups of figures that the root will have one integral and two decimal places. 271. If a number is not a perfect square, ciphers may be annexed, and an approximate value of the root found. Extract to six places of decimals the square root of 19. 19 00 00 00 (4.358899 272. The square root of a common fraction is found by extracting the square roots of the numerator and denomi nator. But, when the denominator is not a perfect square, it is best to reduce the fraction to a decimal and then extract the root. The side of a square is found by extracting the square root of its area. 17. A rectangle is 972 yds. long and 432 yds. wide. Find the side of a square which has the same area as the rectangle. 18. Find in yards the length of the side of a square field containing 27 A. 12 sq. rds. In a right triangle, the square on the hypotenuse (AC) is equal to the sum of the squares on the two legs. Hence hypotenuse = square root of A C sum of squares on the legs; and one leg square root of difference of squares on the other two sides. = 19. Base=39, perpendicular = 52; find hypotenuse. 20. Base 35, hypotenuse 91; find perpendicular. 21. Perpendicular = 72, hypotenuse = 75; find base. 22. A cord 287 ft. long is stretched from the top of a flagpole 63 ft. high; find the distance of the end in contact with the ground from the base of the pole. The length of the diagonal of a room is the square root of the sum of the squares of the length, breadth, and height. 23. Find the diagonal of a room 28 ft. long, 21 ft. wide, and 12 ft. high. 24. Find the diagonal of a hall 50 ft. long, 30 ft wide, and 15 ft. high. CUBE ROOT. 273. The cube of a number is the product of three factors, each equal to the number. are The cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 274. The cube root of a number is one of the three equal factors of the number. are Thus the cube roots of 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 275. The cube root of a number is indicated by, or by the fraction written above and to the right of the number. Thus, 343, or 3434, means the cube root of 343. 30+5, the cube of 35 may be obtained |