Ans. Greatest common measure a-1. Reduced fraction Ans. Common divisor a2-x2. Reduced fraction a2+x2 а-х. 6. Find the greatest common measure, and reduce the fraction a-y to its lowest terms. ay Ans. Common divisor a2-y2. Red. frac. a2+y2 a2+a2y2+y2. 7. Roduce a3-3a2x+3ax2-x3 to its lowest terms. a2-x2 (Art. 28.) We may often reduce a fraction by separating both numerator and denominator into obvious factors, without the formality of finding the greatest common divisor. The following are some examples of the kind: 1. Reduce a3-ab2 a3-ab2 a2+2ab+b2 = to its lowest terms. a(a2-b2) _a(ab)(a+b)_a2-ab a2+2ab+b2 (a+b)(a+b) (a+b)(a+b) a+b.. x5b2x3 x4-64 .2 Reduce to its lowest terms. Ans. x3 x2+b2. 4. Reduce cx+cx2 to its lowest terms. Ans. c+cx ac+ab 5. Reduce acx+abx 2x3-16x6 3x3-24x-9 to its lowest terms. Ans. . (Art. 29.) To find the least common multiple of two or more quantities. The least common multiple of several quantities is the least quantity in which each of them is contained without a remainder. Thus, the least common multiple of the prime factors, a, b, c, x, is obviously their product abcx. Now observe that the same product is the least common multiple also, when either one of these letters appear in more than one of the terms. Take a for example, and let it appear with b, c, or x, or with all of them as a, ab, c, ax, or a, b, ac, ax, the product abcx is still divisible by each of these terms. Therefore, when the same factor appears in any number of the terms, it is only necessary that it should appear once in the product, that is, once in the least common multiple. If it should be used more than once, the product so formed would not be the least common multiple. From this examination, the following rule for finding the least common multiple will be obvious: RULE. Write the given terms, one after another, and draw a line beneath them. Then divide by any prime factor that will divide two or more of the terms without a remainder, bringing down the quotients and the terms that will not divide, to a line below. Divide this second line as the first, forming a third, &c. until nothing but prime quantities are left. Then multiply all the divisors and the remaining terms that will not divide, and their product will be the least common multiple. N. B. This rule is also in common arithmetic. EXAMPLES. 1. Required the least common multiple of Sac, 4a", 12ab, 16ac, and cx. Therefore 2ax2c×2×a×3bxx=24a2cbx. Here the divisor 2c will not divide 2a, but the coefficient of c will divide the coefficient of a, and we let them divide, for it is the same as first dividing by 2, and afterwards by c. From the same consideration we permit 2c to divide cæ, or let the letter c in the divisor strike out c before x. By the rule we should divide by 2 and by c separately, but this is a practical abbreviation of the rule. 2. Required the least common multiple of 72a, 156, Jab, and 3a2. Ans. 135a2b. 3. Find the least common multiple of (a2-x2), 4(а-х), Ans. 4(a2-x2). (a+x). 4 4. Find the least common multiple of ax2, bx, acx, and a2-x2. Ans. (a2-x2)cbx. The least common multiple is useful many times in reducing fractions to their least common denominator. CASE 4. To reduce fractions to a common denominator. (Art. 30.) The rule for this operation, and the principle on which it is founded, is just the same as in common arithmetic, merely the multiplication of numerator and denominator by the same quantity. The object of reducing fractions to a common denominator is to add them, or to take their difference, as different denominations cannot be put into one sum. RULE. Multiply cach numerator by all the denominators, except its own, for a new numerator, and all the denominators for a common denominator. Or, find the least common multiple of the given denominators for a common denominator; then multiply each denominator by such a quantity as will give the common denominator, and multiply each numerator by the same quantity by which its denominator was multiplied. Ans. and 24cdx and 6сх 6сх 6сх |