CHAP. XXXI. OF CUBES, AND THE EXTRACTION OF CUBE ROOTS. 252. We have already seen (Chap. XVI.) that the cube of any number may be found by multiplying that number twice into itself; by the application of which to compound quantities we shall find the cube of a + b, as follows; viz. a + b a+b a2 + ab + ab + b2 a2+2ab+b2 a+b a3 +2ab+ abo + a2b+2ab2 +bs a3+3a2b+3ab2 +b3 This quantity we find to consist of the cubes of each term of the root, viz. a3 and b3, and also of the quantity 3a2b+3ab3, which is equivalent to (a+b).3ab; that is to say, to 3 times the product of the two terms multiplied by their sum. Whenever therefore the root consists of two terms, it is easy by this rule to find the cube of it; for example, the number 53 +2, and the cube of it will therefore be 27+8+(5 x 18)=125. And And if 107 +3, be the root, the cube will be 343 +27+(10 x 63) 1000. To find the cube of 36, let us make the root 30 +6, and we shall then have for the cube, 27000+216 + 36 × 54046656. Now if, on the other hand, the cube be given to find the root, as a3 +3a2b+3ab2+b3, by considering attentively the above examples, we shall be enabled to trace the root of it with very little difficulty. First, if the cube be arranged according to the powers of one letter, we easily know by the first term a3 that a must be the first term of the root, and that the remainder, viz. 3a2b+3ab2+b3 must necessarily therefore furnish the second term of the root. But as we already know that in the instance before us the second term is b, it remains to shew how this second term may be derived from the above remainder. Now this remainder may be divided into two factors, viz. 3a2+3ab+b and b; from which it appears, that if the quantity 3a2b+3ab2+b3 be divided by 3a2 +3ab+b2, we shall obtain b, the second term of the root required. As, however, the second term is supposed to be unknown, the divisor is also unknown: still we have the first term of such divisor, which is sufficient; for it is 3a, that is to say, thrice the square of the first term of the root already found, and by means of it we are enabled to find also the other term b, and then to ascertain completely what the divisor is to be before we perform the division. For this purpose it will be necessary to join to 3a, thrice the product of the two terms of the root, viz. 3ab and b2, or the square of the second term. 253. This 253. This will be better understood by applying it in illustration of the two following examples. a3 +12a2+48a +64 (a, the first term. as In reviewing the last of these examples, it will be seen that when the root consists of more than two terms, the operation must be repeated in the same manner as at first, considering each succeeding term as the second term, or b, and consequently taking for the first, or a, all the preceding terms in the form of compound quantities; as, in order to find the last term of the root in the above example, we assumed a2 - 2a, which are in fact the first and second terms, as the first term a, in order to form a divisor for the last remainder, which divisor will be seen to have been thus obtained. Assuming a2 - 2a in the place of a, we have 3a2=3a* — 12 a3 +12 a2, from which we find the last term of the root to be 1, and consequently 3ab3a2-6a, and b2=1, which being added together, give 3a1-12a3+15a2+1, for the divisor, which being exactly equal to the dividend, shows clearly that 1 is the true third term of the root, which we therefore find to be a2-2a +1. EXAMPLE 3. a3+3a2b+3ab2+b+3a2c+6abc +3b°c + 3 ac2+3bc2 + c3 Divisor, 3a2+6ab+3b2+3ac+3bc+ c2 3a2c+6abc+3b°c + 3 ac2+3 bc2 + c3 (a + b, first term. 3 a°c +6abc+3b°c + 3 ac2+3bc+cc, second term. The analysis which we have given will show the foundation of the rules ordinarily given in books of Arithmetic for the extraction of the cube root in numbers, as will appear from the following examples. |