64. From what is proved in the preceding number, it follows, that any quantity may be transferred from the numerator of a fraction into the denominator, or from the denominator into the numerator, by changing the sign of its index. Thus, ner, a2b3c a3 1 65 = a3× 75 = (since 75 = b−5) a3 b−5. In like man d3 e2x2 = a2b3c5d-3 1 65. A compound quantity, having a fractional index, may be expanded into an infinite series, by means of the Binomial Theorem. For it may be demonstrated, that the general formula stated in No. 50, is equally true, whether n be positive or negative, integral or fractional. The general formula may perhaps be most conveniently expressed in the following form: (a+b)"a"+Pa"-1b+Qa"-2b2+Ra-3b3+Sa-4b1+&c. b b2 b3 64 Or, = an(1+P+Q 2+ R √3+S+ &c.) where in both the a3 a4 a S = R. 3 N -3 " 4 &c. as will appear evident from com paring them with the coefficients of (a+b)", stated in the number already alluded to. The first of these forms is most convenient when n is integral and positive; the second when n is fractional or negative. The application of the Binomial Theorem has already been shown when n is a whole number; let us now endeavour to apply it when n is fractional or negative. Ex. 1. Required the square root of (a+b). That is, required to raise (a+b) to the power whose index is First, To find the coefficients. Here P= 1 2 Q R 3 3 The negative indices of a may be rendered positive by transferring the quantity from the numerator to the de nominator of the fraction, (No. 64.) We shall then have, If the second formula be used, which, in this instance, is This, with every such example, is evidently an infinite series, since none of the factors which compose the coefficients can ever become = = 0. G Ex. 2. Required the cube root of the square of (a2 — x2). This result will obviously be indicated by (a2x2). The coefficients obtained, as in the preceding example, will be, Leaving the coefficients out of consideration, the terms of (a2 — a2) would evidently be + and — alternately; because 2 being a negative quantity, all its odd powers will be negative, and its even powers positive, (No. 48.) If, however, any of the coefficients of these positive terms should come out negative, then we shall have a positive quantity multiplied by a negative; and, consequently, (No. 22.) the product is negative, that is, the term will become negative. Thus, the literal part of the third term, in the above example, is naturally +, the coefficient of it, however, comes out a+ 1 9' and hence the term is In short, if any coefficient, found as before, have the same sign as the term to which it belongs naturally has, that term will be positive; but if it have a different sign, the term will be negative. a = a × (x − y)—} (No. 64.) The coefficients are -- From the preceding example, it appears, that the Binomial Theorem may not only be employed in extracting imperfect roots, but also in finding quotients. Thus, α £(h-x) evidently expresses the quotient of a divided by the cube root of (x-y). If, by means of this theorem, the student find the quotients of a divided by a+x, and of a divided by a-x, or, which is the same thing, expand a(a+x)—1, and a(a-x)-1 into infinite series, he will find the results to agree with those stated on page 68, as found by actual division. The Binomial Theorem may be very conveniently employed for extracting imperfect roots of numbers, more particularly when the number exceeds the complete power, whose root is to be extracted by a very little. fraction) 3(1+.0240817) = 3.0722451 nearly. In all examples of this kind, when the second term of the binomial is small in comparison of the first, it is obvious that a few of the first terms of the expanded form will be sufficiently accurate for most purposes. EXAMPLES. Ex. 1. Convert (a—x)3 into an infinite series. Ex. 2. Convert (a2+x2) into an infinite series. Ex. 3. Required the quotient of a2 divided by the cube root of the square of x2+y2. That is, required to convert into an infinite series. a2 (x2+y2)? Ex. 5. Required the cube root of 67, or of its equal 64+3. SURDS. 66. The definition of a surd has already been given in No. 55. The precise values of surd quantities cannot be ascertained. They may, however, be determined to any degree of exactness less than perfect, by means of decimals or infinite series. In this sense, surds quantities are called irrational, to distinguish them from those, whether integral or fractional, whose values are determinate, and which are therefore called rational. Thus, a+x, 5, are rational quantities; (a+x)31, √5, are irrational. Surds, when properly reduced, are subject to the common elementary rules of Arithmetic and Algebra. 67. A rational quantity may be reduced to the form of a surd, by raising it to the power corresponding to the root of the given surd, and then prefixing the radical sign with the proper index. Ex. 3. Reduce a+x to the form of the square root. a+x=√(a+x)2 or to √a2+2ax+x2. Ex. 4. Reduce 3a to the form of the cube root. sa* =√(3a3) = {√27a. Ans. |