CHAPTER IX. Of Cubes, and the Extraction of Cube Roots. 292. To find the cube of a root a + b, we only multiply its square a a+2ab+bb again by a + b, thus, It contains, therefore, the cubes of the two parts of the root, and beside that, 3 a ab+3abb, a quantity equal to (3 a b) × (a+b); that is, the triple product of the two parts, a and b, multiplied by their sum. 293. So that whenever a root is composed of two terms, it is easy to find its cube by this rule. For example, the number 5 = 3 + 2; its cube is therefore 27+ 8+ 18 × 5 = 125. Let 7310 be the root; the cube will be 343 +27 +63 × 10 = 1000. To find the cube of 36, let us suppose the root 36 = 30 + 6, and we have for the power required, 27000+216 + 540 × 36 = 46656. 294. But if, on the other hand, the cube be given, namely, a3+3aab+3abb + b3, and it be required to find its root, we must premise the following remarks: First, when the cube is arranged according to the powers of one letter, we easily know by the first term a3, the first term a of the root, since the cube of it is a3; if, therefore, we subtract that cube from the cube proposed, we obtain the remainder, 3aab+3abb + b3, which must furnish the second term of the root. 295. But as we already know that the second term is + b, we have principally to discover how it may be derived from the above remainder. Now that remainder may be expressed by two factors, as (3 aa + 3 ab + bb) × (b); if, therefore, we divide by 3aa3ab+bb, we obtain the second part of the root + b, which is required. 296. But as this second term is supposed to be unknown, the divisor also is unknown; nevertheless we have the first term of that divisor, which is sufficient; for, it is 3 a a, that is, thrice the square of the first term already found; and by means of this, it is not difficult to find also the other part, b, and then to complete the divisor before we perform the division, For this purpose, it will be necessary to join to 3 aa thrice the product of the two terms, or 3 ab, and bb, or the square of the second term of the root. 297. Let us apply what we have said to two examples of other given cubes. 298. The analysis which we have given is the foundation of the common rule for the extraction of the cube root in numbers, An example of the operation in the number 2197 : 2197 (10+ 3 = 13 1000 300 1197 90 9 399 1197 0. Let us also extract the cube root of 34965783: 34965783 (300+20 + 7 27000000 270000 7965783 18000 400 288400 5768000 307200 2197783 6720 313969 2197783 0. CHAPTER X. Of the Higher Powers of Compound Quantities. 299. AFTER squares and cubes come higher powers, or powers of a greater number of degrees. They are represented by exponents in the manner which we before explained: we have only to remember, when the root is compound, to inclose it in a parenthesis. Thus (a+b)5 means that a + b is raised to the fifth degree, and (a - b) represents the sixth b. We shall in this chapter explain the nature of these powers. power of a 300. Let ab be the root, or the first power, and the higher powers will be found by multiplication in the following manner : (a+b)1 = a +6 a + b a2 + ab +ab+bb (a+b)2 = a2+2ab+bb a + b a3+2aab+abb +aab+2abb + b3 (a+b)3 = a3+3aab+3abb +63 a+b a*+3a3b+3aabb + ab3 (a + b) = a++4a3b+6aabb +4ab3+b2 a + b a5+4a+b+6a3bb4aab3 +ab+ (a+b)5a5+5a+b+10a3bb+10aab3 +5ab1 +b3 a+b a® +5a3b+10a4bb + 10a3b3 + 5aab1 +ab5 +ab+5a1bb + 10a3b3 + 10aaba +5ab3 +bo (a+b) = a® +6a5b+15a1bb +20a3 b3 + 15aab +6ab3 +bo of the root a powers bare found in the same man 301. The ner, and we shall immediately perceive that they do not differ from the preceding, excepting that the 2d, 4th, 6th, &c. terms are affected by the sign minus; (a —b)1 — a1 —4a3b + 6aabb — 4ab3 +ba. (a—b) — a 5—5ab10a3bb-10aab35ab4 — b3 5 = a5a5b10a4bb — 10a3b3 + 5aab1 — ab 5 a5b+5a1bb +10a3b3 +10aab*—5ab5+bo (a — b) ® — a ® — 6a3b + 15a1bb —20a3b3 + 15aaba— 6ab5+bo Here we see that all the odd powers of b have the sign —, while the even powers retain the sign +. The reason of this is evident; root, the powers of that letter will — b, + bb, — b3, + ba, — b3, +a, &c. which clearly shows that the even powers must be affected by the sign, and the odd ones by the contrary sign 302. An important question occurs in this place; namely, how we may find, without being obliged always to perform the same calculation, all the powers either of a + b, or a — b. We must remark, in the first place, that if we can assign all the powers of ab, those of ab are also found, since we have only to change the signs of the even terms, that is to say, of the second, |