Solution. When the given quantities are decomposed into their simplest factors, as is the case with monomials, their least common multiple is readily obtained; for it is obviously equal to the product of all the unlike factors, each factor being raised to a power equal to the highest power which it has in either of the given quantities.

But the common factors can always be obtained from the process of finding the greatest common divisor.

72. EXAMPLES.

1. Find the least common multiple of 2 a3 b2 c x, 3 a5 b c3 x2, 6 a c x = 2. 3 a c x, 9 c7 x1o — 32 c7 x1o, 24 a3 — 23. 3 a8. Ans. 23.32. a8 b2 c7 x10 =72 a8 b2 c7 x10.

2. Find the least common multiple of 16 a x, 40 b5 x, 25 a7 b3 x2. Ans. 400 a7 b5 x2.

3. Find the least common multiple of ", -1, xn−2, xn−3, x.

-3

Ans. x.

4. Find the least common multiple of 6 (a + b) xm, 54 (a—b)3, (a+b)7, 81 (a—b)3 xm+2, 8(a+b)5 xm−8 ̧

Ans. 648 (a+b)7 (a — b)3 xm+2.

Sum and Difference of Fractions.

5. Find the least common multiple of a2+2 a b + b2, a2 + 4 a b + 4 b2, a2 — b2, a2 + 3 a b + 2 b2, a3 + a2 b

73. Problem. To find the sum or difference of given fractions.

Solution. When the given fractions have the same denominator, their sum or difference is a fraction which has for its denominator the given common denominator, and for its numerator the sum or the difference of the given numerators.

When the given fractions have different denominators, they are to be reduced to a common denominator by arts. 67 and 68.

Sum and Difference of Fractions.

a

5. Reduce to one fraction the expression+c.

75. Corollary. It follows, from examples 3 and 4,

that the sum of half the sum and half the difference

Product and Quotient of Fractions.

of two quantities is equal to the greater of the two quantities; and that the difference of half their sum and half their difference is equal to the smaller of them.

SECTION III.

Multiplication and Division of Fractions.

76. Problem. To find the continued product of several fractions.

Solution. The continued product of given fractions is a fraction the numerator of which is the continued product of the given numerators, and the denominator of which is the continued product of the given denominators.

77. Problem. To divide by a fraction.

Solution. Multiply by the divisor inverted.

The preceding rules for the addition, subtraction, multiplication, and division of fractions require no other demonstrations than those usually given in arithmetic.

78. When the quantities multiplied or divided contain fractional terms, it is generally advisable to reduce them to a single fraction by means of art. 73.

80. The reciprocal of a quantity is the quotient obtained from the division of unity by the quantity.

Hence the product of a quantity by its reciprocal is unity; the reciprocal of a fraction is the fraction inverted; and the reciprocal of the power of a quantity is the same power with its sign reversed.

81. Corollary. To divide by a quantity is the same as to multiply by its reciprocal; and, conversely, to multiply by a quantity is the same as to divide by its reciprocal.