Extraction of the Roots of all Powers. 390. When the index of the power is prime to 2 and 3, and consequently cannot be resolved into those factors, we can no longer obtain the root merely by the square and cube roots, and must, therefore, have recourse to another method. The scholar has already seen that we obtain the formula for extracting the square or cube root by considering the root under the form of a binomial, which binomial we square or cube accordingly. Now in raising a binomial successively to the higher powers, a law is found to exist in the formation of the terms of those powers, by means of which we obtain a general rule or formula for extracting the roots of all powers. Let the binomial be ab, as usual. Then if we raise a+b to any power, we shall find that the first term is a, raised to the given power, and the last term b, raised to the same power. Also, if we subtract a unit from the index of the power, the remainder will show the number of intermediate terms, which consist of ab, the index of a, decreasing, and that of b increasing, each by a unit at a time. The index of b in the second term, being a unit, is not written. The number of terms, therefore, is one greater than the index of the power. Thus, if we would raise a + b to the seventh power, we have, according to the above, the following eight terms: a7, aob, a3b3, als, al, a%b, als, b. 391. To have the number which precedes each of the intermediate terms, called the coefficient, we observe the following rule: The coefficient of the second term is the index of the power. For the coefficient of the third term, multiply the coefficient of the second term by the index of a in that term, and divide the product by 2. For the coefficient of the fourth term, multiply the coefficient of the third term by the index of a in that term, and divide by 3. In the same manner, to find any coefficient, multiply the coefficient of the preceding term by the index of a in that term, and divide by the number of terms to that place inclusive. Thus, for the coefficients of the above series of terms we have for the second term, 7, the index of the power. third (6 7 X 6 =21 The seventh power of a+b is therefore, a7+7ab+ 21a5b2 + 35a1b3 +35a3ba + 21a3b3 + 7ab® + 67. 392. When the power is odd, the number of terms, being one more, is even; and there are two middle terms, which have each the same coefficient. The coefficients which succeed the middle ones are the same as those which precede them, taken in an inverted order. When the power is even, the number of terms is odd, and there is, consequently, but one middle term, the coefficients succeeding which are the same as those which precede it, taken in inverted order. Wherefore, we do not calculate the coefficients farther than the middle one. 393. A general formula, for raising a binomial to any power, according to the above method, is obtained as follows: Suppose we would raise a + b to a power signified by any number Then we have n. This general formula, which was discovered by Sir Isaac Newton, and is therefore called Newton's Binomial Theorem, furnishes a general rule for extracting the roots of all powers. As in the square and cube roots we obtain the divisor from the second term of the formula, in which b first appears in its simplest form, so, from the second term of the general formula, in which b is still found in its simplest form, we obtain the divisor for finding b in any root whatever. In finding the second figure of the root, by means of this divisor, which is na”-1, we are, however, subject to the same kind of difficulty which was experienced in finding the same figure in cube root; and this we shall endeavour to obviate in the same manner as we did in the extraction of that root. Thus, having first divided the second and third terms of the formula by b, to find two terms of a general divisor, we seek, in its simplest form, the ratio of 1 a"-2b to na”-1. To find this, we first multiply both terms by 2, and have n (n − 1) a"-2b and 2na"1. We then divide both by na"-2, and have (n − 1)6 and 2a. ` The must observe, that b cannot exceed 9, and that when, from the formula, we find that the divisor will be considerably increased, it is better to diminish b by at least a unit. 394. To extract the root of any power, therefore, we proceed as follows: As 102-100, 103 1000, 10*: 10000, 105 =100000, &c., it is evident, that in raising to any power, the tens of a binomial root, there will be as many ciphers on the right as there are units in the index of the power; and therefore, to extract any root whatever, it may be shown, as in the square and cube roots, that we must point off the figures of the number into periods of as many figures each as there are units in the index of the This is the first step. power. Seek the greatest figure, which, raised to the given power, is contained in the left-hand period, and place it in the root. This is the second step. This first figure, as well as its re quired power, may easily be found by inspecting the table of powers. Raise the first figure of the root to the given power, subtract the result from the left-hand period, and to the right of the remainder, bring down the left-hand figure of the next period for a new dividend. This is the third step. The reason for bringing down only the left-hand figure is, that as we raise the root found, which is a number of tens, to the next inferior power for a divisor, we should, as in the square and cube root, have to cut off as many figures on the right as there are units in the index of the given power, minus one; that is, as many, minus one, as are contained in a period. Then, for a divisor, raise the figure found in the root to the power next inferior to the given power, and multiply the result by the index of the given power. This is the fourth step. Then in finding the second figure of the root, increase the divisor, as directed above, in order to find b nearer the truth, than by na" alone. This is the fifth step, and very important, seeing that to make several useless trials, in very high powers, is no trifling embarrassment. Next, involve the root to the given power, subtract the result from the two first periods of the given number, and to the right of the remainder bring down the left-hand figure of the next period, to form a new dividend. This may be called the last step. For considering the whole root now found as a, we form the divisor na"-1, and take all the steps in succession as before, except the fifth, which, after the second figure, will not often be needed, observing to subtract the root, when involved, from as many periods as there are figures in the root. او 395. With respect to the extraction of the root of a vulgar or decimal fraction, the scholar can find no difficulty, seeing that the methods of operating, for any root whatever, are exactly analogous to those pursued in the square and cube roots. The limit of the approximation of any root may also be found, as in the cube root. Thus, for the fifth root, the formula being a55a+b+10a3b3 + 10a3b3 + 5aba + b5, and the intermediate terms 5a3b + 10a3b3 + 10a2b3 + 5aba; if we consider b=1, we have, for ascertaining the limit, the formula, 5a +10a3 +10a3 + 5a, or (aa + 2a3 + 2a2 + a)5. Now if, when we have found the root in integers, we involve it, and form a number, according to this formula, 10, 10, 1000, &c. of the number thus formed, will show within what limit we shall find the fifth power of the root, when extracted to one, two, three, &c. places of decimals. Again, for the limit of the fifth root of a fraction, suppose the root to equal a unit, which is evidently too great, then as (a+2a3+2a+a)5 is less than (1+2+2+1)5, that is, less than 30, for one decimal, the limit will be within 38, or 3 units; for two decimals, within; for three, within 180; that is, within. Hence, we shall find the fifth power of the root of a fraction within any limit, by extracting it two places farther than that limit. In a similar manner, the limit may be found for any other root. (n - 1)b 396. The ratio 2a 46 26 is, in this root, or, 26 Now, α if the root contains 2 figures, is less than 2 units; if the root contains three, it is less than; if four, it is less than To; that is, than; consequently, having found a number of figures of the root, we may venture to find, by division, as As a many more, minus three, as we have already found. general rule for all powers, we may always find, by division, as many more figures as we have already found, minus the number sifinified by n-1. 13, we increase 80 by of itself, that is, by 3 × 16=48, and have 128 for the corrected divisor. This divisor gives 4, the true figure of the root, which, in all probability, would not otherwise be found by the student, till after two useless trials. 2. Required the seventh root of 7974586522691918732936169. 7974,5865226,9191873,2936169 (3609 2187 Corrected divisor 78364164096 15237476352000000) 138170113091918732 7974586522691918732936169 In this example, as the divisor 5103 is contained 11 times in 57875, we first assume b=9. Then, as the divisor must, by the formula, be greatly increased, we diminish b by a unit. The ratio (n-1)b 66 2a or 2a 36 is accordingly, a 3 X 8 30 }; hence, we increase the divisor 5103 by of itself, which thus becomes 9183. This divisor gives b 6, which is the true figure of the root. As the second divisor, which is 15237476352, is not contained in the new dividend 13817011309, we place a |