to affect this power with the common radical sign. If the index of the radical sign is divisible by the exponent of the -power in question, the operation then is performed by divid ing that index by the exponent of the power. Let us give two examples for these two cases, (arba)="/apsbbs; (w/arba)="/apbя. ms 326. If the exponent of the power is equal to the index of the radical sign, the power is the quantity under the radical sign. In fact, the indication /ar, shows that a is the mth power of a certain number/a, which we can always assign, either rigorously, or by an approximation, so that the mth power of /ap is ar. In like manner, the square of a is a; the cube of /a is a; the 5th power of 5/(-a2) is—a2 ; and so on. 327. A rational quantity may be reduced to the form of a given surd, by raising it to the power whose root the surd expresses, and prefixing the radical sign. Thus a3/a1=/a® m 10. =/a, &c. and a+x=(a+x)m. In the same manner, the form of any radical quantity may be altered; thus, √(a+x) ={/(a+x)2=∞/(a+x)3, &c. or (a+x)3=(a+x)2=(a+ x) &c. Since the quantities are here raised to certain powers, and the roots of those powers are again taken; therefore the values of the quantities are not altered. Also, the coefficient of a surd may be introduced under the radical sign, by first reducing it to the form of the surd, and then multiplying as in (Art. 318). Thus, axa3× √x=√✓a2x; 6√√36 × √2=√72; and x(2a — x) * = (x2)* × (2a—x)*=√(2ax2. 2.3). 328. Conversely, any quantity may be made the coefficient of a surd, if every part under the sign be divided by this quantity, raised to the power whose root the sign expresses. Thus, ✔(a3 —a2x)=√a3×√(a—x)=a√✓√(a—x); √60=√(4x15)=√ 4X15=215; and (m—amx1)=”/[am×(an~x)]= m/am/(a" -œ”)=α”/ (a1 — x2). n m/n/at. m ገበ 329. Let us pass to the extraction of roots of radical quan tities, and let the mth root of /at be required, which we indicate thus, at. We shall put /ax, or wax, by making atd. Involving both sides to the power m, we find a' or a=rm, raising again to the power n, we obtain at=xmn. If the math root of both sides be extracted, we have another enunciation of x; hamely, mn/at=x="/a2. /V/at= at. And, in fact, we make 1st, ///at=a', whence "/α'=x, and a="/Va'=xm; 2d, by putting at a", whence "a" =xm, and a"=xmn; 3d, making Vata", whence 1/a"=xm2, mnpq and a”=\/a2=xmnp; and finally a2=xmnpq, ..x= √ at. Thus, for example, the 12th root of the number a can be transformed into //a. m 1 330. If, in the equality a=a", where a is supposed to represent a number greater than unity, we make m= we shall have ed to P 卫 0 a=a". Let now go, and we shall be conduct P a=a2 =a=1: Now is equal to infinity, (Art. 165,) or it is the superior limit of numbers; therefore unity is the limit of the roots whose index continually increases. If po, we have a=a. Therefore, from the index zero to infinity, the root passes from infinity to unity. 331. To the hypothesis pq, corresponds p a=a=a" =a. So that, in passing from the index 1 to the index zero, the root runs over the digression of numbers, from the given number inclusively to infinity. And, finally, let us suppose that p=0, and q=0 ; then aa =a, which is an indeterminate quantity; since the expo 0 nent is the mark of indetermination (Art. 201). Ο 332. It is to be observed, that radical quantities or surds, when properly reduced, are subject to all the ordinary rules of arithmetic. This is what appears evident from the preceding considerations. It may be likewise remarked, that, in the calculation of surds, fractional exponents are frequently more convenient than radical signs. § 11. REDUCTION of radical quANTITIES OR SURDS. CASE I. To reduce a rational quantity to the form of a given Surd. RULE. 333. Involve the given quantity to the power whose root the surd expresses; and over this power place the radical sign, or proper exponent, and it will be of the form required. Ex. 1. Reduce a to the form of the cube root. Here, the given quantity a raised to the third power is a3, and prefixing the sign, or placing the fractional exponent () over it, we have a=3/a3=(a3)3 (Art. 312). 334. A rational coefficient may, in like manner, be reduced to the form of the surd to which it is joined; by raising it to the power denoted by the index of the radical sign. Ex. 2. Let 5a= √25×√a=√25a (Art. 317). Ex. 3. Reduce -3a2b to the form of the cube root. Here, (-3a2b)—27ab3 ; :: −3/27a63 is the surd required. Ex. 4. Reduce -4xy to the form of the square root. Ex. 5. Reducer to the form of the cube root. Ans. (1). Ex. 6. Reduce a+z to the form of the square root. Ans. (a2+2az+22). Ex. 7. Reduce 4x4 to the form of the cube root. Ans. (2/64x3) or (64x3)3. Ex. 8. Reduce -X y to the form of the square root. Ans. √xy. Ex. 9. Reduce ab to the form of the square root. CASE II. Ans.ab. To reduce Surds of different indices to other equivalent ones, having a common index. RULE. 335. Reduce the indices of the given quantities to fractions having a common denominator, and involve each of them to the power denoted by its numerator; then 1 set over the common denominator will form the common index. Or, if the common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. Then over the said quantities, with their new indices, set the given index, and they will make the equivalent quantities sought. Ex. 1. Reduce u and b to surds of the same radical sign. Here, ✓ a=a3, and b=63. Now, the fractions and reduced to the least common denominator, are and ; ‚·‚a‡=a3=(a3)' =∞/a3, and b'3=b*=(b2)*=&/b2 Consequently a3 and 63 are the surds required, Ex. 2. Reducea and to surds of the same radical sign, or to the common index . (Art. 312), √a=a3, and √/x=x*; then ÷÷J=1×6=3; and 1÷1=1×6=§; .. √/a3 and √/xa, or (a3)* and (x3)*, are the quantities required. Ex. 3. Reduce a and b to the same radical sign 3⁄4/. Ans. Vao, and b Ex. 4. Reduce a and x3 to surds of the same radical sign. Ans. 12/a3 and 1/1. Ex. 5. Reduce a and m/y to surds of the same radical sign. Ans. mam and my". Ex. 6. Reduce a and b to surds of the same radical sign. Ans. 15/a5 and 1/63. mn Ex. 7. Reduce 33/2 and 25 to the same radical sign. Ans. 3/4 and 2/125. Ex. 8. Reduce /xy and Vax to the same radical sign. Ans. /xy and 12/a3x3. CASE III. To reduce radical Quantities or Surds, to their most simple forms. RULE. 336. Resolve the given number, or quantity, under the radical sign, if possible, into two factors, so that one of them may be a perfect power; then extract the root of that powr, and prefix it, as a coefficient to the irrational part. Ex. 1. Reduce ab to its most simple form. ท m Here ar=/a" X/x=a" Xx=axx. 337. When the radical quantity has a rational coefficient prefixed to it; that coefficient must be multiplied by the root of the factor above mentioned; and then proceed as before. Ex. 4. Reduce 53/24 to its simplest form. Here 53/24=52/(8×3)=53/8X3/3=5×2×3/3-103/3. Ex. 7. Reduce(a3+a3b2) to its most simple form. Âns. a3/(1+b2). Ex. 9. Reduce (a+b)3⁄4/[(a—b)3Xx2] to its most simple form. Ans. (a2-b2)3/x2. 338. If the quantity under the radical sign be a fraction, it may be reduced to a whole quantity, thus: Multiply both the numerator and denominator by such a quantity as will make the denominator a complete power corresponding to the root; then extract the root of the fraction whose numerator and denominator are complete powers, and take it from under the radical sign. a2 Ex. 1. Reduce X to an integral surd in its most sim ple form. 쿨 (718)=X/18=/18. 2 X 32 33 Ex. 3. Reduce to an integral surd in its most simple form. Ans. 14 |