Law of Signs in Multiplication. By the definition of multiplication (§ 78), since and +3+1 + 1 + 1, 3x (+8)= +8 +8 +8 = + 24, (3) × 8-8-8-8-24, : and (−3) × (− 8) = − (−8) − (− 8) − (− 8) = = +8+8+8 = +24. The minus sign before the multiplier, 3, signifies that the repetitions of the multiplicand are to be subtracted. If a and b stand for any two numbers, we have (+ a) × (+ b) = + ab, That is, if two numbers have like signs, the product has the plus sign; if unlike signs, the product has the minus sign. 80. The Law of Signs in Multiplication is, therefore, Like signs give +, and unlike signs give The Index Law in Multiplication. a2 = Since = aa, and a3 = aaa, a2 x a3 = a2+3; a4+1. If then m and n are positive integers, am xan = am+n In like manner, am × a" × ao = am+n+p. 81. The Index Law in Multiplication is, therefore, The exponent of a letter in the product is equal to the sum of the exponents of the letter in the factors of the product. Multiplication of Monomials. 1. Find the product of 6 a2b3 and 7 ab2c3. Since the order of the factors is immaterial, 6 a2b8 × 7 ab2c3 = 6 × 7 × a2 × a × b3 × b2 x c3 2. Find the product of 3 ab and 7 ab3. - 3 ab × 7 abs = 3 x 7 x ax a xbx b3 3. Find the product of x" and x3, and of x and x”. xnx x8 = x2 +з. xn xxnxn+ n = x2n. Therefore, 82. To Find the Product of Two Monomials, ($ 42) Find the product of the numerical coefficients; and to this product annex the letters, giving to each letter an exponent equal to the sum of its exponents in the factors. 83. A product of three or more factors is called the continued product of the factors. 1. Find the continued product of (-a)×(—b)×(−c). By the law of signs, § 80, we have (a) (b) = ab, and (ab) × (−c) = -abc. 2. Find the continued product of (− a) × (—b) × (− c) × (− d). 84. From Examples 1 and 2 (§ 83), we see that an odd number of negative factors gives a negative product; and an even number of negative factors gives a positive product. EXERCISE 14. NOTE. The beginner should first write the sign of the product; then the product of the numerical coefficients after the sign; and, lastly, the letters in alphabetical order, giving to each letter the proper exponent. Find mentally the product of: Division of Algebraic Numbers. 85. Division is the operation of finding one of two factors, when their product and the other factor are given. 86. With reference to this operation the product is called the dividend, the given factor the divisor, and the required factor the quotient. Law of Signs in Division. =- Since (+ a) × (+b) = + ab, ·· + ab ÷ (+ a) = + b. .. — ab ÷ (+ a) = − b. + b. − b. That is, if the dividend and divisor have like signs, the quotient has the plus sign; and if they have unlike signs, the quotient has the minus sign. 87. The Law of Signs in Division is, therefore, Like signs give +, and unlike signs give — The Index Law in Division. The quotient contains the factors of the dividend that are not found in the divisor. If m and n are positive integers, and m is greater than n, am ÷ an = am-n 88. The Index Law in Division is, therefore, The exponent of a letter in the quotient is equal to the exponent of the letter in the dividend minus the exponent of the letter in the divisor. Division of Monomials. 1. Divide 24 a' by 8 a5. 24 a7 = 3 a7-5 = 3 a2. We obtain the factor 3 of the quotient by dividing 24 by 8; and the factor a2 of the quotient, by writing a with an exponent equal to the exponent of a in the dividend minus the exponent of a in the divisor. 2. Divide 20 a5b6 by — 4 ab3. bn bn bn bn NOTE. Since by division 1; and by the index law = bo, it follows that b = 1. Hence, any letter which by the rule would appear in the quotient with zero for an exponent, may be omitted without affecting the quotient. 89. To Find the Quotient of Two Monomials, therefore, Divide the numerical coefficient of the dividend by the numerical coefficient of the divisor; and to the result annex the letters, giving to each letter an exponent equal to its exponent in the dividend minus its exponent in the divisor, |