155. Multiply their numerators together for a new numerator, and their denominators together for a new denominator; reduce the resulting fraction to its lowest terms, and it will be the product of the fractions required. It has been already observed, (Art. 119), that when a fraction is to be multiplied by a whole quantity, the numerator is multiplied by that quantity, and the denominator is retained: a ac 2x and Thus, xc=' 10x -×5=- ; or, which is the same, b making an improper fraction of the integral quantity, and a C ac b' then proceeding according to the rule, we have X=' Ђ Hence, if a fraction be multiplied by its denominator, the α ab. product is the numerator; thus, X b = = b. b In like manner, the result being the same, whether the numerator be multiplied by a whole quantity, or the denominator divid ed by it, the latter method is to be preferred, when the denominator is some multiple of the multiplier; Thus, let ad ad be the fraction, and e the multiplier; then Xc= bc adc bc Also, when the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity which is common to each of them, the quotients may be used instead of the fractions themselves; thus, Ꮖ ; cancelling a+b in the numerator of the aba+b=a a-b one, and denominator of the other. Here, (3x+2)x8x=24x2+16xz= numerator, and 4×7=28= denominator; 24x2+16x the product required. Therefore, 28 by 3 a a2-x2 7x2 Here, (a2-x2) × 7x2=(a+x)× (a−x) × 7x2= numerator (Art. 106), and 3a× (a—x)= denominator; see Ex. 15, (Art. 79). Hence, the product is (a+x) x (a-x) × 722 the numerator and denominator by a-x), (dividing 7x2(a + x) = 3a Then, (5a+2)X(3a-x)=15a2-2ax-x2- new numerator, and 5x3 15 denominator: Therefore, 15a2-2ax-x2 15 156. But, when mixed quantities are to be multiplied together, it is sometimes more convenient to proceed, as in the multiplication of integral quantities, without reducing them to improper fractions. Ex. 9. It is required to find the continual product of 3a 2x2 a+b and 5' 3 1 ax 2ax+2bx Ans. 5 Ex. 10. It is required to find the continued product of a2x2 a+y a2 — y21 a2+x2 27 Ex. 11. It is required to find the continued product of a2x2 a2-b2 and α a+b'a+x2 ax-x2 a2-ab Ans. Ex. 12. Multiply 2-3x+1 by x2-1x. To divide one fractional quantity by another. RULE. 157. Multiply the dividend by the reciprocal of the divisor, or which is the same, invert the divisor, and proceed, in every respect, as in multiplication of algebraic fractions; and the product thus found will be the quotient required. When a fraction is to be divided by an integral quantity; the process is the reverse of that in multiplication; or, which is the same, multiply the denominator by the integral, (Art. 120), or divide the numerator by it. The latter mode is to be preferred, when the numerator is a multiple of the divisor. a2 - b2 a-b -a-b; hence is the fraction required. Ꮖ ; hence a+b X = Ꮖ a+b a+b 158. But it is, however, frequently more simple in practice to divide mixed quantities by one another, without reducing them to improper fractions, as in division of integral quantities, especially when the division would terminate. 8 - Ex. 5. Divide -+- 1x by x2-x. x2-x)x-4x+11x2-4x(x2-4x+1 3 INVOLVING ONLY ONE UNKNOWN QUANTITY. 159. In addition to what has been already said, (Art. 34), it may be here observed, that the expression, in algebraic symbols, of two equivalent phrases contained in the enunciation of a question, is called an equation, which, as has been remarked by GARNIER, differs from an equality, in this, that the first comprehends an unknown quantity combined with certain known quantities; whereas the second takes place but between quantities that are known. Thus, the expression S d α= +5, (Art. 102), according to the above remark, is called 2' an equality; because the quantities a, s, and d, are supposed to be known. And the expression x+x-d-s, (Art. 103), is called an equation, because the unknown quantity x, is combined with the given quantities d and s. Also, x-a=0 is an 2 |