From the above process we derive the following rule : Separate the number into periods by pointing every second figure, beginning with the units' place. Find the greatest square in the left-hand period, and write its square root as the first figure of the root; subtract its square from the number, and to the result bring down the next period. Divide this remainder, omitting the last figure, by twice the part of the root already found, and annex the quotient to the root and also to the divisor. Multiply the complete divisor by the figure of the root last obtained, and subtract the product from the remainder. If other periods remain, proceed as before, doubling the part of the root already found for the next trial-divisor. Note 1. It should be observed that decimals require to be pointed to the right. Note 2. As the trial-divisor is an incomplete divisor, it is sometimes found that after completion it gives a product greater than the remainder. In such a case, the last root-figure is too large, and one less must be substituted for it. Note 3. If any root-figure is 0, annex 0 to the trial-divisor, and bring down to the remainder the next period. EXAMPLES. 207. 1. Find the square root of 49.449024. 49.449024 | 7.032, Ans. 49 1403 4490 4209 14062 28124 Since the second root-figure is 0, we annex 0 to the trialdivisor 14, and bring down to the remainder the next period, 90. If there is a final remainder, the given number has no exact square root; but we may continue the operation by annexing periods of ciphers, and thus obtain an approximate value of the square root, correct to any desired number of decimal places. 14. Extract the square root of 12 to five figures. 12.00000000 | 3.4641..., Ans. 9 64 300 256 686 4400 4116 6924 28400 27696 69281 70400 69281 1119 Extract the square roots of the following to five figures : The square root of a fraction may be obtained by taking the square roots of its terms. If the denominator is not a perfect square, it is better to reduce the fraction to an equivalent fraction whose denominator is a perfect square. Thus, to obtain the square root of as follows: 3 8 , we should proceed Extract the square roots of the following to five figures : 208. Since (a+b)3 = a3+3a2b+3ab2 + b2, we know that the cube root of a3+3a2b+3ab2 + b2 is a + b. It is required to find a process by which, when the quantity a3+3a2b+3ab2 + b is given, its cube root, a + b, may be determined. The cube root of the first term is a, which is the first term of the root. Subtracting its cube from the given expression, the remainder is 3a2b+3ab2 + b2, or (3a2 + 3ab + b2) b. Dividing the first term of this remainder by 3a2, or three times the square of the first term of the root, we obtain b, the second term. Adding to the trial-divisor 3ab, that is, three times the product of the first term of the root by the second, and 62, that is, the square of the last term of the root, completes the divisor, 3a2 + 3ab + b2. This being multiplied by b, and the product, 3a2b+3ab2 + b3, subtracted from the remainder, completes the operation. From the above process we derive the following rule: Arrange the terms according to the powers of some letter. Find the cube root of the first term, write it as the first term of the root, and subtract its cube from the given expression. Divide the first term of the remainder by three times the square of the first term of the root, and write the quotient as the next term of the root. Add to the trial-divisor three times the product of the first term of the root by the second, and the square of the second term. Multiply the complete divisor by the term of the root last obtained, and subtract the product from the remainder. If other terms remain, proceed as before, taking three times the square of the root already found for the next trialdivisor. EXAMPLES. 209. 1. Find the cube root of 8x-36x2y + 54x2y2 -27 y3. 8x-36x+y+54x2y2 - 27 y3 | 2x2-3y, Ans 8x6 12x2-18x2y+9y2-36x2y + 54x2y2 - 27 y3 - 36 x2y + 54x2y2 - 27 y3 The cube root of the first term is 2x2, which is the first term of the root. Subtracting 8x6 from the given expression, we have - 36 xty as the first term of the remainder. Dividing this by three times the square of the first term of the root, 12x4, we obtain - 3y as the second term of the root. Adding to the trial-divisor three times the product of the first term of the root by the second, - 18 x2y, and the square of the second term, 9y2, completes the divisor, 12 x 18 x2y + 9 y2. Multiplying this by -3y, and subtracting the product from the remainder, there is no remainder. Hence, 2 x2 - 3y is the required cube root. 2. Find the cube root of 40x3 - 6x5 - 96x + x - 64. The second complete divisor is formed as follows: The trial-divisor is 3 times the square of the root already found; that is, 3 (x2-2x)2, or 3x4 - 12x2 + 12x2. Three times the product of the root already found by the last term of the root is 3(-4) (x2-2x), or 12x2 + 24 x; and the square of the last root-term is 16. Adding these, we have for the complete divisor 3x4 – 12 x3 + 24 x + 16. Find the cube roots of the following: 3.1-6y+12y2 - 8 y3. 4. 27x+27x2 + 9x2+1. 5. 54 ху2+27 y3 + 36 x2y +8x3. 6. 64 – 144 a2xy + 108 ax2y2 - 27 xxy. 7. x + 6x5 - 40x3 + 96 x — 64. 8. y-1+5y – 3 y – 3 у. 9. 15x4 6x - 6x + 15 x2 + 1 + x – 20 x. 10. 9x - 21 x2 - 36 x + 8x-9x + 42x4 -1. |