According to the preceding Article. the result must be 1 But, according to Art. 68, may be written a3 ; 1 That is, the Rule of Art. 50 is general, and applies to nega tive as well as positive exponents. Ex. 2. Multiply -b- by b-3. 3. Multiply a by a'. 4. Multiply b3 by b3. 7. Multiply (a-b)' by (a—b)—3. Ans. -b. PROBLEM VIII. (91.) To divide one fractional quantity by another. RULE. Invert the divisor, and proceed as in multiplication. If the two fractions have the same denominator, then the quotient of the fractions will be the same as the quotient of their numerators. But when the two fractions have not the same denominator. we must reduce them to this form by Problem IV. ad Reducing to a common denominator, we have to be di bd bc vided by bd It is now plain that the quotient must be represented by the division of ad by bc, which gives ad bc the same result as obtained by the above Rule. According to the Rule of the preceding Article, we have 1 a3 a3 1 = That is, the Rule of Art. 66 is general, and applies to nega tive as well as positive exponents. Ex. 2. Divide -b- by -b-. (93.) According to the definition, Art. 33, the reciprocal of a quantity is the quotient arising from dividing a unit by that quantity. That is, the reciprocal of a fraction is the fraction inverted. Hence, 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. (94.) The numerator or denominator of a fraction may be itself a fraction; Such expressions are easily reduced by applying the preceding principles. SECTION VII. SIMPLE EQUATIONS. (95.) An equation is a proposition which declares the equality of two quantities expressed algebraically. Thus, x-4-b-x, is a proposition expressing the equality of the quantities x-4 and b-x. The quantity on the left side of the sign of equality is called the first member of the equation; the quantity on the right, the second member. Equations are usually composed of certain quantities which are known, and others which are unknown. The known quantities are represented either by numbers or by the first letters of the alphabet, a, b, c, &c. ; the unknown quantities by the last letters, x, y, z, &c. An identical equation is one in which the two members are identical, or may be reduced to identity by performing the operations which are indicated in them. A root of an equation is the value of the unknown quantity in the equation. (96.) Equations are divided into degrees, according to the highest power of the unknown quantity which they contain. Those which contain only the first power of the unknown quantity are called simple equations, or equations of the first degree. Those in which the highest power of the unknown quantity |