the sign expresses. Thus, √(a3-a2x)=√ a2 × √ (a−x} =a√(a-x); √60=√(4×15)=√4X√15=2/15; and */ (am1-aTMx") =”/[aTM× (a”—x")]="/a®×TM/(an— x")=a*/(a”—x”). mn 329. Let us pass to the extraction of roots of radical quantities, and let the mth root of wat be required, which we indicate thus, a'. We shall putax, or a=x, by making at a'. Involving both sides to the power m, we find a' or at =x, raising again to the power n, we obtain a2= . If the mnth root of both sides be extracted, we have another enunciation of x; namely, We shall find, by a like calculation, mnpq α. And, in fact, we make 1st, Vata', whence "a" =x, and a'=~//Ya'=xTM; 2d, by putting a2= a", whence "a"x", and a"=xm; 3d, making Vat a", whence a"xm", and a""=at=xmp; and finally at=xmnpq, .*.x= mnpq at. Thus, for example, the 12th root of the number a can be transformed into VV. 330. If, in the equality "/a=a", where a is supposed to represent a number greater than unity, we make m=2, we shall have yaa. Let now q=0. ૧. P and we shall be conducted to ✔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. ૧ If p=o, we have Va=a. Therefore, from the index zero to infinity, the root passes from infinity to unity. va=va=a2=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 po, and q=0; P then a a=a", which is an indeterminate quantity ; since the exponent is the mark of indetermination (Art. 201). 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 calculaiion of surds, fractional exponents are frequently more convenient than radical signs. § II. 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=2/a3= (a3)* (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 ; .. —/27ab3 is the surd required. Ex. 4. Reduce -4xy to the form of the square root. Here, (-4xy)=16x2y2 ; .'. (Art. 116), —4xy ==√16x2y2. Ex. 5. Reduce x to the form of the cube root. Ans. (†x3)3 ̧ Ex. 6. Reduce a+z to the form of the square root. 14 Ans. (a2+2az+z2)*. Ex. 7. Reduce 4x4 to the form of the cube root. I 3 Ans. (3/64x3) or (64x3)3. Ex. 8. Reducexy to the form of the square Ans.√xy. root. Ex. 9. Reduce ab to the form of the square root. CASE II. 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 in dices 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 radical sign. a and b to surds of the same Herc,a=a3, and 66. Now, the fractions and reduced to the least common denominator, are & and } ; .•. a2=a*=(a3)*=÷/a3, and b*=b3 = (ba)*=2/63. Consequently quired. Ex. 2. Reduce I radical sign, or to the common index t. 4 (Art. 312), √/a=a3, and/x=x; then ÷÷÷÷×6 =3; and÷×6=}; .../a3 and 3 and (x), are the quantities required. I 3 3, or (a3)* Ex. 3. Reduce a and b to the same radical sign Ex. 4. Reduce a and to surds of the same Ans. 12/a3 and 12x. radical sign. Ex. 5. Reduce a and my to surds of the same Ex. 7. Reduce 33/2 and 2/5 to the same radical sign sign. Ans. 3/4 and 25/125. ax to the same radical 12 Ans. 1/x'y' and 1a3 3. 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 power, and prefix it, as a coefficient, to the irrational part. Ex. 1. Reduce ab to its most simple form. Ex. 2, Reduce /ame to its most simple form. m m m m Here /ax/a" ×"/x=a"> x=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 52/24-53/(3x3)=53/8×3/3=5×2×3/ 3=103/3. Ex. 5. Reduce abc and 98ax to their most simple form. Ans. abc and 7a/2x. Ex. 6. Reduce /243 and 5/96 to their most simple form. 3 Ans. 3/3 and 2/3. Ex. 7. Reduce (a3+asb2) to its most simple form. Ans. a3/(1+63). Ex. 8. Reduce √ (ab-4a*b*+4ab3) to its -26 Ans. ✔ab. C |