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 37 and to equivalent surds, having a common And 3 x 4 12 common denominator. Then 3=3×3×3×3×3×3× 3 × 3|17=6561 ans. Also 29|TT=2×2× 2 × 2 × 2 × 2 × 2×2×2}=512π,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 29, making 3' and 2o ri 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. Reduce x and y to equal surds, having a common index. First 1x3=3 new index of x. 1x2=2........ of y. 2x3=6 common denominator. Wherefore x and y are the surds required. 11. Reduce x+y) and x-y to a common index. Thus 1×3=3 index of x+y. 2x3=6 common denominator. Therefore (x+y)3}+=x3+3x2y+3 xy2+ylt, ans. The reason of this rule will appear from the 9th example; for andre duced to a common denominator, become respectively and; wherefore 3 becomes 31 and 24 becomes 217: but 3 8th power, or 37=6561|17; and 27° implies the 12th root of 3 in the implies in like manner 2o 1'r= 512, as in the example, and the like of other quantities; wherefore the rule is manifest. 12. Reduce 7 and 8 to a common index. and 4096. Ans. 117649 13. Reduce x, y and z to a common index. Ans. x3\t'TM, y+', and 'T. 14. Reduce a and b to a common index. Ans. aand mu, 15. Reduce a+x and z to a common index. 4 Ans, a+4a3x+6a2x2 + 4 ax3 + x * ) and zlo, 62. To reduce surds to equivalent ones having a given index. RULE 1. 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. OPERATION. Wherefore 24+ and 3++=81 are the quantities required. Explanation. I first divide and each by 4, 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 4 for the answer. I I 17. Reduce a and z to equal surds, having 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 4/24 √/2*=*√/22 = 14, and 3÷ is in like manner evidently equal to 4/3 all which is sufficiently plain from Art. 60. 44 4 = 4 4 Wherefore (a) and (2), the answer. 18. Reduce 2 and 3 to equal surds, having the common 1 19. Reduce x+y and 34 to equal surds, having for a com 2 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 1 S and 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=√16x2=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 4/2 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 /i6 × 2: 16 x 2, but √16=4, wherefore √16X √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. 3 22. Reduce 108 rty to its simplest terms. 3 108x+y=√27 a3 y3 × 4 xy2=3 ay3 /4xy'. Ans. 23. Reduce 8 x3-12 x2y to its simplest terms. 24. Reduce √/8x3-12x2y=√√/4x2 x2x-3y=2x√2x-3y. Ans. 25. Reduce 26. Reduce 3/4 a3xy and 3416x5 to their simplest terms. Ans. ay3/4 xy and 12 x1 √x. 27. Reduce 3x3-x+y+32x+z3 3 5 and 3 1000 x7 to their simplest terms. Ans. x* √√3.x-y+32 z2 and 6 x3 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 '. 3 • To prove the truth of this rule, let a3-be a given surd, of which it is 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 bc2 be multiplied by c2, and it will become a3- ; 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 3c3; 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 the surd part is reduced to an integer, and the whole 3 bc2 is of the same с ~응, value with the given surd a3√, which was to be shewn, First, the numerator 1 being multiplied by its denominator 3, 2 produces 3 for the surd; then, dividing the coefficient by 3, 3 Multiplying the numerator 4 by 5, gives 20 for the surd; then 3 whence 20/20 is the surd, which reduced to its simplest terms Multiply 2a by (the square of 5x or) 25 x2, and the product 50 ax' will be the surd part; then divide the coefficient 1 (under 1 stood) by 5x, and the quotient is the coefficient: wherefore 5.x 50 147 3 3 31. Reduce √: and to equivalent surds, having the 65. ADDITION OF SURDS. RULE I. Reduce the quantities to a common index, the |