and m-n; we may therefore conclude that, the number of different combinations of m letters taken n and n, is equal to the number of combinations of m letters taken m -n and m - n. For example, twelve letters combined 5 and 5, give the same number of combinations as when taken 12-5 and 12-5, or 7 and 7. Five letters combined 2 and 2, give the same number of combinations as when combined 5 2 and 5 -2, or 3 and 3. 208. If, in the general formula, That is, the sum of the co-efficients of all the terms of the for mula for the binomial, is equal to the mth power of 2. Thus, in the particular case (x + a)5 = x + 5ax2 + 10a2x2 + 10a3x2 + 5a*x + a5, the sum of the co-efficients 1+5+10+10+5+1 is equal to 25 = 32. In the 10th power developed, the sum of the co-efficients is equal to 210 = 1024. Extraction of the Cube Root of Numbers. 209. The cube or third power of a number, is the product which arises from multiplying the number twice by itself. The cube root, or third root of a number is either of three equal factors into which it may be resolved; and hence, to extract the cube root, is to seek one of these factors. Every number which can be resolved into three equal factors that are commensurable with unity, is called a perfect cube; and any number which cannot be so resolved, is called an imperfect cube. The first ten numbers are roots, 1, 2, 3, 4, 5, 6, cubes, 8, 9, 10; 7, Reciprocally, the numbers of the first line are the cube roots of the numbers of the second We perceive, by inspection, that there are but nine perfect cubes among all the numbers expressed by one, two, and three figures. Every other number, except the nine written above, which can be expressed by one, two, or three figures, will be an imperfect cube; and hence, its cube root will be expressed by a whole number, plus an irrational number, as may be shown by a course of reasoning entirely similar to that pursued in the latter part of Art. 118. 210. Let us find the difference between the cubes of two consecutive numbers. Let a and a + 1, be two consecutive whole numbers; we have whence, (a + 1)3 = a3 + 3a2 + 3a + 1; That is, the difference between the cubes of two consecutive whole numbers, is equal to three times the square of the least number, plus three times the number, plus 1. Thus, the difference between the cube of 90 and the cube of 89, is equal to 3 (89)2 + 3 x 89 + 1 = 24031. 211. In order to extract the cube root of an entire number, we will observe, that when the figures expressing the number do not exceed three, the entire part of the root is found by comparing the number with the first nine perfect cubes. For example, the cube root of 27 is 3. The cube root of 30 is 3, plus an irrational number, less than unity. The cube root of 72 is 4, plus an irrational number less than unity, since 72 lies between the perfect cubes 64 and 125. When the number is expressed by more than three figures, the process will be as follows. Let the proposed number be 103823. This number being comprised between 1,000, which is the cube of 10, and 1,000,000, which is the cube of 100, its root will be expressed by two figures, or by tens and units. Denoting the tens by a, and the units by b, we have (Art. 198), (a + b)3 = a3+3a2b+3ab2 +63. Whence it follows, that the cube of a number composed of tens and units, is made up of four distinct parts: viz., the cube of the tens, three times the product of the square of the tens by the units, three times the product of the tens by the square of the units, and the cube of the units. Now, the cube of the tens, giving at least, thousands, the last three figures to the right cannot form a part of it: the cube of tens must therefore be found in the part 103 which is separated from the last three figures. The root of the greatest cube contained in 103 being 4, this is the number of tens in the required root. Indeed, 103823 is evidently comprised between (40)3 or 64,000, and (50)3 or 125,000; hence, the required root is composed of 4 tens, plus a certain number of units less than ten. Having found the number of tens, subtract its cube, 64, from 103, and there remains 39, to which bring down the part 823, and we have 39823, which contains three times the square of the tens by the units, plus the two parts named above. Now, as the square of tens gives at least hundreds, it follows that three times the square of the tens by the units, must be found in the part 398, to the left of 23, which is separated from it by a point. Therefore, dividing 398 by 48, which is three times the square of the tens, the quotient 8 will be the units of the root, or something greater, since 398 hundreds is composed of three times the square of the tens by the units, together with the two other parts. We may ascertain whether the figure 8 is too great, by forming from the 4 tens and 8 units the three parts which enter into 39823; but it is much easier to cube 48, as has been done in the above table. Now, the cube of 48 is 110592, which is greater than 103823; therefore, 8 is too great. By cubing 47 we obtain 103823; hence the proposed number is a perfect cube, and 47 is its cube root. REMARK I. The units figures could not be first obtained, because the cube of the units might give tens, and even hundreds, and the tens and hundreds would be confounded with those which arise from other parts of the cube. REMARK II. The operations in the last example have been performed on but two periods. It is plain, however, that the same reasoning is equally applicable to larger numbers; for, by changing the order of the units, we do not change the relation in which they stand to each other. Thus, in the number 43 725 658, the two periods 43 725, have the same relation to each other, as in the number 43725; and hence, the methods pursued in the last example are equally ap plicable to larger numbers. 212. Hence, for the extraction of the cube root of numbers, we have the following RULE. I. Separate the given number into periods of three figures each beginning at the right hand: the left-hand period will often contain less than three places of figures. II. Seek the greatest cube in the first period, at the left, and set its root on the right, after the manner of a quotient in division. Subtract the cube of this figure of the root from the first period, and to the remainder bring down the first figure of the next period, and call this number the dividend. III. Take three times the square of the root just found for a divisor, and see how often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the first two periods of the given number, diminish the last figure; but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period, for a new dividend. IV. Take three times the square of the whole root for a new divisor, and seek how often it is contained in the new dividend; the quotient will be the third figure of the root. Cube the whole root, and subtract the result from the first three periods of the given number, and proceed in a similar way for all the periods. REMARK. If any of the remainders are equal to, or exceed, three times the square of the root obtained plus three times this root, plus one, the last figure of the root is too small and must be augmented by at least unity (Art. 210). To extract the nth Root of a whole Number. 213. The nth root of a number, is one of the n equal factors into which the number may be resolved. If the factors are com mensurable with unity, the number is said to be a perfect power, if they are not commensurable with unity, the number is said to be an imperfect power. In order to generalize the process for the extraction of roots, we will denote the proposed number by N, and the degree of the root to be extracted by n. If the number of figures in N, does not exceed n, the rational part of the root will be expressed by a single figure. Having formed the nth power of all the numbers from 1 to 9, inclusive, compare the given number with these powers, and the root of the one next less, will be that part of the required root which can be expressed by a whole number; for, the nth power of 9 is the largest number which can be expressed by n figures. When N contains more than n figures, there will be more than one figure in the root, which may then be considered as composed of tens and units. Designating the tens by a, and the units by b, we have (Art. 203), N = (a + b) = a" + nan-16+ n n-1 an-262 +, &c.; that is, the proposed number contains the nth power of the tens, plus n times the product of the n-1th power of the tens by the units, plus a series of other parts which it is not necessary to consider. Now, as the nth power of the tens, cannot give units of an order inferior to 1 followed by n ciphers, the last n figures on the right, cannot make a part of it. They must then be pointed off |