Va=√ a = 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 p=0, and q=0; then a P a°, which is an indeterminate quantity; since the exponent (Art. 201). It 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. 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=/a3=(a3)? (Art. 312). 3 334. 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)3——27ab3 ; .'. —/27ab3 is the surd required. Ex. 4. Reduce -4xy to the form of the square root. Here, (-4xy)=16x3y2; .. (Art. 116), -4xy =-16x2 y2. Ex. 5. Reduce x to the form of the cube root. I Ans. (x3)3. Ex. 6. Reduce a+z to the form of the square root. 1 Ans. (a2+2az+z2)2. Ex. 7. Reduce 4x4 to the form of the cube root. Ex. 8. Reduce root. I -xy to the form of the square Ans. Ex. 9. Reduce ab to the form of the square root. Ans. -a2b3. 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 a and b to surds of the same radical sign. Here, ✓ a=a3, and bb. Now, the fractions and reduced to the least common denominator, a*=(a3)*='/a3, and b*=b*= Ex. 2. Reduce a and to surds of the same radical sign, or to the common index . I I 4 (Art. 312), a=a3, and 1/x=x^; then 3 2 Ꮖ =3; and ÷÷÷={×6=}; .../a3 and √x3, or (a3)* and (x3), are the quantities required. Ex. 3. Reduce a and b to the same radical sign 3 Ex. 4. Reduce a and to surds of the same radical sign. x Ex. 5. Reduce a and Wy to surds of the same Ex. 7. Reduce 3/2 and 2/5 to the same radical sign Ans. 3/4 and 2/125. Ex. 8. Reducery and //ax to the same radical sign. Ans. 12xy' and 'ax 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 a2b to its most simple form. m Here /ax=/a" ×"/x=amx/x=αxx. Ex. 3. Reduce 72 to its most simple form. Here ✓72 (36×2)=36x/2=6/2. 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 53/(8×3)=53/8×3/3=5×2×3/ 3=103/3. Ex. 5. Reduce a'bc and 98a2x to their most simple form. Ans. abc and 7a/2x. Ex. 6. Reduce 4/243 and 5/96 to their most simple form. Ans. 3/3 and 25/3, Ex. 7. Reduce (a3+a3b2) to its most simple form. Ans. a/(1+b). (a3b-4a2b2+4ab3· Ex. 8. Reduce √(a most simple form. to its -26 Ans. Vab. Ex. 9. Reduce (a+b)3⁄4/[(a−b)3 Xx2] to its most simple form. Ans. (a2 —b23/)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. vb. Ex. 2. Reduce to an integral surd in its simplest form. 2X32 33 27 X 3 -=32/(4 × 18)=4×13/18=2/18. Ex. 3. Reduce to an integral surd in its most simple form. to integral surds Ans. by and ca3. 2 y to integral surds in Ans.27 and Ex. 5. Reduce and their most simple form. 2. to their most sim Ans. 2 and |