Find the square root of 3; of 357.357. Find to four decimal places the square root of: 236. Since the cube of a + b is a3+3a2b+3ab2 + b3, It is required to devise a method for extracting the cube root, a + b, when a3+3a2b + 3ab2 + b3 is given : the cube root of a3+3a2b+3ab2 + b2 is a + b. The first term, a, of the root is obviously the cube root of the first term, a3, of the given expression. If a3 is subtracted, the remainder is 3 a2b + 3ab2 + b2; therefore, the second term, b, of the root is obtained by dividing the first term of this remainder by three times the square of a. Also, since 3a2b + 3ab2 + b2 = (3 a2 + 3ab+b2) b, the complete divisor is obtained by adding 3 ab + 2 to the trial divisor 3 a2. Find the cube root of 8 x3 + 36 x2y + 54 xy2 + 27 y3. 8 x3 + 36 x2y + 54 xy2 + 27 y3 2 x + 3 y 8 x3 (6x+3y)3y = 12 x2 36 x2y + 54 ху2 + 27 y3 The cube root of the first term is 2x, and 2x is therefore the first term of the root. 8x3, the cube of 2x, is subtracted. The second term of the root, 3 y, is obtained by dividing 36 x2y by 3 (2x)2 = 12 x2, which corresponds to 3 a2 in the typical form, and the divisor is completed by annexing to 12 x2 the expression {3 (2x) + 3 y} 3 y = 18 xy + 9 y2. 237. The same method may be applied to longer expressions by considering a in the typical form 3 a2 +3 ab + b2 to represent at each stage of the process the part of the root already found. Thus, if the part of the root already found is x + y, then 3 a2 of the typical form will be represented by 3(x + y)2; and if the third term of the root is + z, the 3 ab + b2 will be represented by 3(x + y) z + z2. So that the complete divisor, 3 a2 + 3 ab + b2, will be represented by 3(x + y)2 + 3(x + y) z + z2. Find the cube root of x − 3 x5 + 5x8 – 3x - 1. The root is placed above the given expression for convenience of arrangement on the page. The first term of the root, x2, is obtained by taking the cube root of the first term of the given expression; and the first trial divisor, 3x4, is obtained by taking three times the square of this term. The first complete divisor is found by annexing to the trial divisor (3 x2 - x) (— х), which expression corresponds to (3a + b) b in the typical form. The part of the root already found (a) is now represented by x2 - x; therefore, 3 a2 is represented by 3 (x2 - x)2 = 3 x4 - 6 x3 + 3x2, the second trial divisor; and (3a + b)b by (3x2 — 3 х — 1) (— 1); therefore, in the second complete divisor, 3a2 + (3a + b) b is represented by (3 x4 6 x3 + 3x2) + (3 x2 – 3x – 1) (— 1) = 3 x4 —- 6 x3 + 3x + 1. EXERCISE 87. Find the cube root of: 1. a3+3a2x + 3 ax2 + x8. 2.8+12x + 6 x2 + x3. 3. x - 3 ax5 + 5 a3x3 — 3 ах — аo. 4.1-6x+ 21 x2 44 x3 + 63 x4 5.1-3x + 6 x2 7 x3 + 6 x − 3 x5 + x6. 6. x + 1- 6x - 6x5 + 15 x2 + 15 x4 20 x8. 54 x5 + 27 x6. 7. 64 x6 - 144 x5 + 8 - 36 x + 102 x2 171 x2 + 204 x4. 10. 27+108 x + 90 x2 - 80 x3 – 60 x2 + 48 x5 - 8 x6. 238. Arithmetical Cube Roots. In extracting the cube root of an arithmetical number, the first step is to arrange the figures in groups. Since 1 = 18, 1000 = 103, 1,000,000 = 1008, and so on, it follows that the cube root of any number between 1 and 1000, that is, of any number which has one, two, or three figures, is a number of one figure; and that the cube root of any number between 1000 and 1,000,000, that is, of any number which has four, five, or six figures, is a number of two figures; and so on. If, therefore, an integral cube number is divided into groups of three figures each, from right to left, the number of figures in the root will be equal to the number of groups. The last group to the left may have one, two, or three figures. 239. If the cube root of a number has decimal places, the number itself will have three times as many. Thus, if 0.11 is the cube root of a number, the number is 0.11 × 0.11 × 0.11 = 0.001331. Hence, if a given number contains a decimal, we divide the number into groups of three 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 three figures, annexing ciphers when necessary. 240. Notice that if a denotes the first term, and b the second term of the root, the first complete divisor is 3a2+3ab + b2, and the second trial divisor is 3 (a + b)2, that is, 3a2+6 ab +362, which may be obtained by adding to the preceding complete divisor its second term and twice its third term. Extract the cube root of 5 to five places of decimals. After the first two figures of the root are found, the next trial divisor is obtained by bringing down the sum of the 210 and 49 obtained in completing the preceding divisor; then adding the three numbers connected by the brace, and annexing two ciphers to the result. The last two figures of the root are found by division. The rule in such cases is that two less than the number of figures already obtained may be found by division without error, the divisor being three times the square of the part of the root already found. |