Imaginary quantities may, however, be subjected to the operations described in the preceding rules: thus, v=axv=b= vab, which is a real quantity. v-8a2b-v4a2 x-2b2av—2b. 3 OF FRACTIONAL EXPONENTS. (64.) We have hitherto expressed irrational quantities by placing them under the radical sign, but they may also be written with fractional exponents, a method which, as has been shown (45), is deduced from the rule for extracting the roots of monomials. By this system of notation, va is written a, va2 is written a3, and, in general, any root is expressed by a fractional exponent, which has, for its numerator, the exponent of the given quantity, and for its denominator, the index of the required root. This method of notation is, on some accounts, more convenient than the other, for the rules of multiplication, division, &c., are the same for these as for monomials with integral exponents. MULTIPLICATION OF QUANTITIES HAVING FRACTIONAL EXPONENTS: RULE. (65.) Multiply the coefficients together, and add the exponents of the same letters: (66.) Divide the coefficient of the dividend by the coefficient of the divisor, and subtract the exponents of the same letter. 1. Divide a3 by a EXAMPLES. Ans. a3 + a1 — a†(−1) — a}+} _ að. = EXTRACTION OF ROOTS. RULE. (68.) Extract the root of the coefficient, and divide the exponent of each letter by the index of the required root. Thus, the square root of as is a; and the cube root of a is a ++3 = a+. EXAMPLES. a POLYNOMIALS HAVING RADICAL TERMS. (69.) The same rules will apply to polynomials, some, or all the terms of which are radicals, as have been applied to polynomials in general, with this difference only, that the operations performed upon the radical terms of these polynomials, follow the rules which have been laid down in the preceding pages. It will be sufficient, therefore, to give a number of examples, for exercise, in the application of those rules. |