120. Multiplication of Quantities affected with Negative Expo nents.—Suppose it is required to multiply by 1 a2 According to the preceding article, the result must be 1 But, according to Art. 76, 73 a2 may be 1 written a-2; and may be written a-3. a5 Hence we see that a-3xa-2-a-b; that is, the rule of Art. 58 is general, and applies to negative as 121. If the two fractions have the same denominator, then the quotient of the fractions will be the same as the quotient of their numerators. Thus it is plain that is contained in as often as 3 is contained in 9. If the two fractions have not the same denominator, we may perform the division after having first reduced them to a common denominator. Let it be required to divide by 2. с Reducing to a common denominator, we have ad to be di bd bc vided by It is now plain that the quotient must be reprebd sented by the division of ad by bc, which gives. ad; a result which might have been obtained by inverting the terms of the divisor and multiplying by the resulting frac tion; that is, a C α xd=ad bc db Hence we have the following RULE. Invert the terms of the divisor, and multiply the dividend by the resulting fraction. Entire and mixed quantities should first be reduced to frac tional forms. 122. Division of Quantities affected with Negative Exponents. Suppose it is required to divide preceding article, we have 1 by by a3 According to the that is, the rule of Art. 72 is general, and applies to negative as well as positive exponents. EXAMPLES. 1 Ans. -a-3, or 1. Divide a-5 by —a-2. 2. Divide a2 by a-1. 3. Divide 1 by a-4. 5. Divide bm-n by bm. 6. Divide 12x-2y-4 by -4xy2. 7. Divide (x-y)—1 by (x—y)—∞. 123. The Reciprocal of a Fraction.-According to the definition in Art. 34, the reciprocal of a quantity is the quotient arising from dividing a unit by that quantity. Hence the reciprocal of is b b α α that is, the reciprocal of a fraction is the fraction inverted. a b+x Thus the reciprocal of is ; and the reciprocal of 1 b+c is b+c. b+x α It is obvious that to divide by any quantity is the same as to multiply by its reciprocal, and to multiply by any quantity is the same as to divide by its reciprocal. 124. How to simplify Fractional Expressions.-The numerator or denominator of a fraction may be itself a fraction or a mixed 21 quantity, as In such cases we may regard the quantity 4 above the line as a dividend, and the quantity below it as a divisor, and proceed according to Art. 121. The most complex fractions may be simplified by the application of similar principles. |