If the NOTE.-In comparing the reduced with the original forms of such expressions as these, the learner will bear in mind the remarks in the NOTE at page 152. saving of arithmetical labour in the actual extraction of the surd root is the object, this object will not always be attained by changing the original form, although such a change may simplify it in so far as it will reduce the surd-factor to the smallest integral value possible. As already observed, the change spoken of is more likely to contribute to arithmetical facility when the quantity under the radical is a fraction, than when it is an integer; but even fractional surds are sometimes of less convenient calculation in the simplified than in their original form; for instance, in example 3, solved above, it is easy to see that, except for particular numerical values of a and b, it is less troublesome to divide ab by a—b, and to extract the square root of the quotient, than to take the squares of a and b, to extract the square root of the difference of those squares, and then to divide the result by a-b. But if a table of square roots be used, preference would, in general, be given to the changed form; in the original form, we should have to take from the table √(a+b) and √(a—b), and then to divide the former by the latter; in the changed form, we should have to take √(a2-b2) from the table, and to divide this tabular number by a-b. 68. To reduce two or more surds to others with a common index. This reduction consists in merely bringing the indices to a common denominator; hence the rule is as follows: RULE.-Use the notation of exponents instead of radical signs; bring the exponents to a common denominator; perform the involution indicated by the numerator, and prefix to the result the radical sign with the denominator for index, or else write over it the equivalent exponent. Thus taking the general expressions /a", b, the surd index n in the one is different from the surd index s in the other; the radical sign being removed, the proper exponents are which are surds having the common surd-index ns. It is un necessary to give particular examples here; the use of the reduction will be exemplified in the operations of multiplication and division. 69. ADDITION AND SUBTRACTION OF SURDS. RULE. Reduce the surds to their most simple forms (p. 151); then if the surd part be the same in all, add or subtract the coefficients of the surds, and annex the common surd to the result. If the surd parts, after the reduction, be different, the addition or subtraction can only be indicated. NOTE: expressions which have a common surd part, that is, the same common surd factor, are called similar surds; surds not reducible to this similarity are called dissimilar surds. (1) Add together the surds 72 and 32. Here ✓72=√(36×2)=6√2; and √32=√(16×2)=4√2; ..√72+√32=6√2+4√2=102. (2) Subtract 2 from 3%. 192(64 × 3)=4/3; √/24= √(8 × 3) = 2/3; 3/32=3(8×4)=6/4; ../192+24+332=43/3+23/3+63⁄4=6(3/3+3/4). (5) 7x3/2α2x2+5a3/2a2x5—3ax3/2α2x2. (6) √ a3b+ ab2-bab. (7) 48abc-a^/ bc2. (13) 8-2 +3√27. (14) 6/4a2+22a. (15) 48-318-5√2+72-√15+2√ √ NOTE.-In comparing the reduced with the original forms of such expressions as these, the learner will bear in mind the remarks in the NOTE at page 152. If the saving of arithmetical labour in the actual extraction of the surd root is the object, this object will not always be attained by changing the original form, although such a change may simplify it in so far as it will reduce the surd-factor to the smallest integral value possible. As already observed, the change spoken of is more likely to contribute to arithmetical facility when the quantity under the radical is a fraction, than when it is an integer; but even fractional surds are sometimes of less convenient calculation in the simplified than in their original form; for instance, in example 3, solved above, it is easy to see that, except for particular numerical values of a and b, it is less troublesome to divide a+b by a-b, and to extract the square root of the quotient, than to take the squares of a and b, to extract the square root of the difference of those squares, and then to divide the result by a-b. But if a table of square roots be used, preference would, in general, be given to the changed form; in the original form, we should have to take from the table √(a+b) and √(a—b), and then to divide the former by the latter; in the changed form, we should have to take √(a2-b2) from the table, and to divide this tabular number by a-b. 68. To reduce two or more surds to others with a common index. This reduction consists in merely bringing the indices to a common denominator; hence the rule is as follows: RULE. Use the notation of exponents instead of radical signs; bring the exponents to a common denominator; perform the involution indicated by the numerator, and prefix to the result the radical sign with the denominator for index, or else write over it the equivalent exponent. Thus taking the general expressions a", b, the surd index n in the one is different from the surd index s in the other; the radical sign being removed, the proper exponents are ms and ; these, in a common denominator, are and m n hence a"a"s =/ams, or ՊՏ nr ns which are surds having the common surd-index ns. It is un necessary to give particular examples here; the use of the reduction will be exemplified in the operations of multiplication and division. 69. ADDITION AND SUBTRACTION OF SURDS. RULE. Reduce the surds to their most simple forms (p. 151); then if the surd part be the same in all, add or subtract the coefficients of the surds, and annex the common surd to the result. If the surd parts, after the reduction, be different, the addition or subtraction can only be indicated. NOTE: expressions which have a common surd part, that is, the same common surd factor, are called similar surds; surds not reducible to this similarity are called dissimilar surds. (1) Add together the surds 72 and √ 32. Here ✓72=√(36×2)=6√2; and √32=√(16x2)=4√2; ..√72+√32=6√2+4√2=10√2. (2) Subtract 2 from 3. .. 3√3-2√16=√10—√10=√10. (3) Required the sum of 192, 3/24 and 3/32. ../192+24+332=43/3+23/3+634=6(3/3+34). (5) 7x3/2α2x2+5a3/2α2x2-3ax/2a2x2. (6) va3b+ ab2-bab. (7) 48abc-a/ be2. (15) 48-318-5√2+√72-√15+2√√. 155 RULE. Reduce the surds to equivalent ones expressing the same root (153), and they will then be prepared for multiplication or division in the ordinary way. (1) Multiply 28 by 33/16. Here the exponents are and, which in a common denominator are and ; ..√8x168 × 16=(8 x 16") = √(8 x 8' x 16') = (8 x 26 x 4 x 2o)=(32 x 4°)=4/32; ..28×3/16=24/32. (2) Divide 12 by 24. The exponents here also are & and : 12# 123 ,33 X 26 = =3. 32 x 26 (3) Divide 3/2-5/2 by/3. Here the surds 32 and 3 require no preparation: the surds 5/2 and 3, when reduced to a common index, are 5(2) and 3. Hence the operations we have to perform are 3√2÷/3, and 5(2)÷3. 6 32 Now, 3√3√√6; and 54÷V/33=5&/ (9) (√6+√2—√1⁄2)×√6. (10) (10/18+1√8)×√ 2. - (11) (√18—2/45)÷√3. (12) 2××16÷32. (13) (ay)*(bx)* (a'y2)3. NOTE.-It may perhaps seem to the learner that this article on the multiplication and division of surds is rendered superfluous by the general treatment of these ope |