If aa and b are both divided by a' the value of the expression remains Whence the product of a fractional by an integer expression is obtained by either multiplying the numerator or dividing the denominator of the fraction by the integer. Again, in the formula a a' aa! a aa' = = let b=1; then ax fb • b' Consequently an integer is multiplied by a fractional expression by multiplying the integer by the numerator of the fraction, and dividing the product by its denominator. From these formula is deduced the following rule for the multiplication of algebraic expressions of the fractional form: 1st. To multiply an integer by a fractional expression: multiply the integer expression by the numerator, and divide the product by the denominator of the fractional expression. 2d. To multiply a fractional by an integer expression: multiply the numerator or divide the denominator of the fractional multiplicand by the integer multiplier. 3d. To multiply one fractional expression by another: multiply together the numerators of the fractional factors for the numerator of the product, and their denominators for its denominator. Examples of the multiplication of algebraic fractions: The quotient of one fraction by another is, consequently, obtained by multiplying the dividend by the reciprocal of the divisor. In the formula a α' a b then %+d=bd ba"" a ba' If a and ba are both divided by a' the value of the fraction is not Whence, the dividend being a fractional and the divisor an integer expression, the quotient is obtained by either multiplying the denominator or dividing the numerator of the dividend by the divisor. Whence, when the dividend is an integer and the divisor a fractional expression, the quotient is obtained by multiplying the dividend by the denominator of the divisor, and dividing the product by its numerator. From these cases is derived the following rule for division when either the dividend or the divisor, or both the dividend and the divisor, are fractional expressions: 1st. To divide a fractional by an integer expression: multiply the denominator or divide the numerator of the dividend by the divisor. 2d. To divide an integer by a fractional expression: multiply the dividend by the denominator of the divisor, and divide the product by its numerator. 3d. To divide one fractional expression by another: multiply the dividend by the reciprocal of the divisor. Examples of the division of algebraic fractions: 36. The product of two powers of the same root is obtained by affecting that root with an exponent equal to the sum of the exponents of the multiplicand and multiplier (Part I. Art. 75), and the quotient arising from the division, one by the other, of two powers of the same root, by affecting that root with an exponent equal to the excess of the exponent of the dividend over the exponent of the divisor (Part I. Art. 100). In applying this rule to multiplication the numerical values of the exponents of the multiplicand and multiplier do not necessarily enter into consideration; but in its application to division the case is otherwise. It is in effect assumed that the dividend (which may be denoted by a") is the product of the divisor (a") by some other power (as a2) of the root a, and that m is greater than n. It may, however, happen This expression, a", is new, and requires explanation. Since in every division in which the dividend is equal to the divisor the quotient is 1, therefore Whence a=1. In the same manner it can be proved that the value of any other root affected with the exponent 0 is 1. It is therefore agreed by convention to regard the expression a', or any quantity having zero for exponent, as equal to unity. The expression of a quantity raised to a power equal to zero owes its existence to the rule of exponents, and not to that of division; for it is evident that the result obtained by the rule of division is 1, since the quotient of any quantity divided by itself is 1. In the division of one monomial by another, and also in the process of simplifying a fraction, any quantity (such as a") which is common to the dividend and the divisor disappears from the quotient or the simplified fraction. Sometimes it is important to preserve a quantity from disappearing, and in such a case the expression a' is employed. 2d. Let n be greater than m, and equal to m+p. 1 ap a −(m+r) =a ̄". The expressions and a being obtained, the first by the common process of division, and, the second by a rule derived from that process, are regarded as equivalent; and, in general, every quantity affected with a negative (or subtractive) exponent is considered as equivalent to the quotient of i divided by that quantity, with the sign of its exponent changed. Consequently any quantity may be removed from the denominator to the numerator of a fraction by changing the sign of its exponent. 37. It is sometimes convenient to employ quantities which have subtractive exponents; and therefore it becomes necessary to determine in what manner quantities with such exponents are to be combined, in any calculation, with each other, and with quantities whose exponents are additive. Now, if it is required to multiply a" by a-", it follows, from Article 36, that Whence the rules for the multiplication and division of quantities which have exponents of which some are additive and others subtractive, or of which all are subtractive, are comprehended (with the interpolation of the word "algebraic") in the rules for the multiplication and division of quantities with additive exponents; namely, that the exponent of the product of two powers of the same root is the algebraic sum of the exponents of the factors, and the exponent of the quotient of two powers of the same root the algebraic difference between the exponents of the dividend and the divisor. 38. a" is the expression of the letter a written m times as a multiplier; a represents the same letter, a, written m times as a divisor; and a indicates that a is the same number of times a factor of the dividend and of the divisor. Since zero, multiplied by itself any number of times, gives a product equal to zero, it follows that Effecting the division of zero by zero, any number whatever may be assumed as quotient; for any number multiplied by zero gives for product zero, which is in this case the dividend. The expression O' appears, therefore, to involve an indefinite number of numerical values; but the expression has in many cases a fixed and determinate value, Thus the fraction Aa thus, Aa", and making a=0, Aa"-" becomes Ax0-", In the case of m>n, or m=n+d, Ax0"-" becomes AX Cd=0, |