Thus, 23=2×2×2=8. Then S, the answer. And 3x=3x × 3 x × 3 x × 3 x=81 x1, and 81 x1, the ans. 3. Reduce to the form of the square root. Thus, a})2=a*=at. Thena, the answer. 4. Reduce a-x to the form of the square root. Thus, a—¿] 2=a2-2 ax+x2; then a2-2 ax+x2, the ans. 5. Reduce 7 to the form of the square root, and a to the form of the cube root. Ans. 49 and 3 60. To put the coefficient of a surd under the radical sign. RULE I. Reduce the coefficient to the form of the given surd, by the preceding rule. II. Multiply the quantities under the radical sign by the reduced coefficient, and the product will be the answer o. 6. Given 2√x, in order to put the coefficient under the radical sign. First reducing 2 to the form of the square root, we have 2= √4, then multiplying x by 4, and placing the common radical sign over the product, we have 4x for the answer. 7. Introduce the coefficients of 33a and 25a-x under the radical sign. Thus, 3=3×3×34=3 √27, then 3/27a, the answer. 5 And 2=2×2×2×2×2 = 3√32; then 5√32.a—x, 5√32a-32x, the answer. or 8. Given 5/2, 3 a3 √5 x, xy* √/3, and a* √2+x-z, to place the coefficients under the radical sign. Ans. 50, 135 a3x, 61. To reduce dissimilar surds to equivalent ones, having a common index. RULE I. Reduce the indices to fractions having a common denominator, and place each new numerator over the quantity to which it belongs, for a new index. • This rule is equally obvious with the preceding; for it is evident that 2 X √x=√4× √x=√4x, as in the 6th example. Let x=9, then 2x= √ 4× √9= √4×9= √36=6=2 × 3; whence 2 √x= √4x. II. Write 1 over the common denominator, and place this fraction as an index over the given quantities, with their new indices; if those quantities are numbers, involve them to the power denoted by the new index, over which place the said fraction for an index P. 9. Reduce 31 and 2 to equivalent surds, having a common index. Then 3=3×3×3×3×3×3×3×5TT=65611' ans. Also 29=2×2×2×2×2×2×2×2×2}+=5127, ans. Explanation. I first reduce and to a common denominator, I place the numerator 8 over the quantity 3, and the numerator 9 over 2; then putting 1 for a numerator over 12, I place this fraction over 38 and 2o, making 3 and 2o Tri I then involve 3 to the 8th power, and 2 to the 9th; over each of which I put the index for the answers. 10. Reducer and y' to equal surds, having a common index. 2x3=6 common denominator. Wherefore x and y are the surds required. 11. Reduce y and x- y to a common index. Thus 1x3=3 index of x+y. 2x3=6 common denominator. Therefore (x+y)3)=x3+3x2y+3xy2+y, ans. P The reason of this rule will appear from the 9th example; for and reduced to a common denominator, become respectively, and; wherefore 33 becomes 3 and 24 becomes 217: but 31 implies the 12th root of 3 in the 8th power, or 3=65617; and 27 implies in like manner 2'T= 512 as in the example, and the like of other quantities; wherefore the rule is manifest. 9 12. Reduce 7a and Sto a common index. and 4096. Ans. 117649+ 13. Reducer, y and z to a common index. y', and 'T. 14. Reduce a and b to a common index. mu. 15. Reduce a+ and z to a common index. 2 Ans. a*+4a3x+6a2x2 +4 ax3 +x+)+ and z 1. Ans. x3, Ans. am and 62. To reduce surds to equivalent ones having a given index. RULE I. Divide the indices of the surds by the given index, and place the quotients each over its proper quantity, for a new index. II. Over the given quantities with their new indices, write the given index, the results will be the surds required ". 16. Reduce 2 and 3 to equal surds, having the common index 4. OPERATION. 4 First =2 the index of 2. the index of 3. Wherefore 24+ and 3+4=81 quired. Explanation. are the quantities re I first divide and each by, and the quotients are 2 and; these I place as new indices respectively over 2 and 3, and over these the given index I then involve 2 to the square, and 3 to the 4th power, by which the new index of the former is taken away, and that of the latter reduced to; over the results I place the given index for the answer. 17. Reduce am and za to equal surds, having index. 1 T 1 S mr. 9 Dividing the indices is equivalent to extracting the root denoted by the divisor; and placing the divisor as an index over the result is equivalent to involving the result to the power denoted by the divisor: wherefore since equal evolution and involution take place, the value of the given quantity is not altered by the transformation effected under this rule; this is evident from Ex. 16. where 2 is evidently equal to 4 4 √27 √/2# = ˆ √/22 =√14, and 34 is in like manner evidently equal to 4 √/3+==+ √33=* all which is sufficiently plain from Art. 60. =4 Wherefore (a) and (r), the answer. 18. Reduce 23 and 3 to equal surds, having the common 19. Reduce x+y and 34 to equal surds, having mon index. Ans. x2+2xy + y2+ and 3+. 20. Reduce x, y and z to equivalent quantities, having s for a common index. Ans. x, y, and zi. S 63. To reduce surds to their simplest terms. RULE I. Divide the given surd by the greatest power (of the same name with it) that will divide it without remainder, and place the said power and the quotient, with the sign between them, under the radical sign. II. Extract the root of the fore-mentioned power, and place its root before the said quotient, with the proper radical sign between them". 21. Reduce 32 to its simplest terms. OPERATION. √32=√16×2=4/2, the answer. Explanation. The greatest square that will divide 32 is 16, and the quotient is 2; the root of 16 is 4, therefore 42 is the answer. In this rule the given surd is resolved into two factors, one of which is a power of the same name with the surd. Now it is evidently the same thing to multiply the remaining factor by this power, both being under the radical sign, or to multiply the factor under the sign by the root of the power not under the sign; thus, Ex. 21. ✓ 32 is evidently equal to √16 × 2= √16 × √2, but √16=4, wherefore √16× √2=4 × √2=4/2, as in the example; wherefore, the transformation which takes place under this rule does not alter the value of the given surd, which was to be shewn. 22. Reduce 3/108 aty to its simplest terms.? 3 √103 x1y3=3 √√/27 a3y3 × 4 xy2 = 3 ay3 √√4xy2. Ans. 23. Reduce /8 x3-12 x2y to its simplest terms. √/8 x3 — 12x2 y = √√/4x2 × 2x−3y=2x√√2x−3y. Ans. 24. Reduce 50 to its simplest terms. Ans. 5 √2. 25. Reduce 24 to its simplest terms. Ans. 2 √6. 26. Reduce 3/4 a3xy1 and 3a √√/16 x3 to their simplest terms. Ans. ay3 4xy and 12 x*x. 27. Reduce 3x-x1y+32x+z3] 3 and 3 1000x to their 5 3 simplest terms. Ans. x3.x-y+32 z2 and 6 x2 3 √x. 64. To reduce a fractional surd to its equivalent integral one. RULE I. Multiply the numerator under the radical sign by that power of its denominator, whose index is 1 less than the index of the surd. II. Take the denominator away from under the radical sign, and divide the coefficient by it, and the surd part will be an integral quantity, which must be reduced to its simplest terms by the foregoing rule '. value of the given quantity. Now it is evident, that if both terms of a fraction be multiplied by the same quantity, (let that quantity be whatever it may,) the value of the fraction is not altered; wherefore let both terms of the given surd be multiplied by c2, and it will become a bc2 ; now the denominator c3 is a complete power of the same denomination with the surd, and therefore (Art. 63.) it may be taken away, provided its root c be made the divisor of the coefficient a; (for dividing by c is the same as dividing by 3/c3; and dividing either one factor, the other factor, or the product, by the same or equal quantities, produces in each case the same result;) wherefore the given surd b C bc2 c3 a becomes successively a3 √ and -3bc2; in the latter of which a 3 the surd part is reduced to an integer, and the whole√be? is of the same value with the given surd a3 which was to be shewn, |