165. The process of addition of certain forms of fractions is simplified by application of the principles governing the signs of fractions (Art. 152). Changing the form of the second fraction, in order that the terms of the factors in all the denominations may be in the same order, we have (Art. 152): Changing the signs of both factors of the denominator of the second fraction, we have changed the signs of an even number of factors, and the sign of the fraction is not changed. Changing the sign of one factor in the denominator of the third fraction changes the sign of the fraction. CHAPTER XIII FRACTIONS (Continued) MULTIPLICATION. DIVISION. THE COMPLEX FORM MULTIPLICATION OF FRACTIONS (a) A FRACTION MULTIPLIED BY A FRACTION The product of two fractions is obtained as follows: 166. The product of two fractions is a fraction whose numerator is the product of the given numerators, and whose denominator is the product of the given denominators. (b) A FRACTION MULTIPLIED By an Integer Since any integral expression, b, has a denominator, 1: That is: 167. A fraction is multiplied by an integral expression if its numerator is multiplied by that expression. The process of multiplication of fractions is simplified if factors common to numerators and denominators of the given fractions are canceled before multiplication. The cancellations are: 8 in 16, 5 in 25 and 15, a in a3, c in c2, m2 in m3, and x1 in x1. Writing each fraction with numerators and denominators factored, m+ 1 (m − 1)2 m2-4 m+1 (m-1) (m-1) X m-1 m2 - 3 m + 2 m2-1 Canceling common factors = m-1 (m−1)(m−2) (m+2)(m-2) (m+1)(m-1) In applying cancellation select factors for divisors from the numerator only. Begin at the left of the numerator and seek a possible cancellation for each new factor considered. |