1 1 3. Reduce a2 and as to quantities that shall have a common index. 1 1 5. Reduce an and om to quantities that shall have a common index. To reduce surds to their most simple forms. RULE-Resolve the given number, or quantity, into two factors, one of which shall be the greatest power contained in it, and set the root of this power before the remaining part, with the proper radical sign between them.* 1. Let 2. Let EXAMPLES. 48 be reduced to its most simple form. Here √ 48 = √(16 x 3) = 4√3. Ans. Note 1. When any number, or quantity, is prefixed to the surd, that quantity must be multiplied by the root of the factor above mentioned, and the product be then joined to the other part, as before. EXAMPLES. 1. Let 232 be reduced to its most simple form. Here 232 = 2√(16 x 2) = 8√2. Ans. 2. Let 53 24 be reduced to its most simple form. Here 53 24 = 53 (8 x 3) = 103/3. Ans. Note 2. A fractional surd may also be reduced to a more convenient form, by multiplying both the numerator and denominator by such a number, or quantity, as will make the denominator a complete power of the kind required; and then joining its root, with 1 put over it, as a numerator, to the other part of the surd.t * When the given surd contains no factor that is an exact power of the kind required, it is already in its most simple form. Thus, 5 nor 3, is a square. 15 cannot be reduced lower, because neither of its factors, + The utility of reducing surds to their most simple forms, in order to have the answer in decimals, will be readily perceived from considering 2 EXAMPLES. be reduced to its most simple form. ==√(×14)=√14. Ans. 2 49 49 2. Let 3 be reduced to its most simple form. 3. Let 125 be reduced to its most simple form. Ans. 55 4. Let√294 be reduced to its most simple form. Ans. 76. 5. Let 56 be reduced to its most simple form. 7. Let 7 80 be reduced to its most simple form. 8. Let 93 81 be reduced to its most simple form. Ans. 23/7. 6. Let / 192 be reduced to its most simple form. Ans. 43. Ans. 285. Ans. 7a√2x· 12. Let√(x3- a2x2) be reduced to its most simple form. Ans. x√(x - a2). the first question above given, where it is found that = 14; in which case it is only necessary to extract the square root of the whole number 14, (or to find it in some of the tables that have been calculated for this purpose,) and then divide it by 7; whereas, otherwise, we must have first divided the numerator by the denominator, and then have found the root of the quotient, for the surd part; or else have determined the root both of the numerator and denominator, and then divided one by the other; which are each of them troublesome processes when performed by the common rules; and in the next example for the cube root, the labour would be much greater. CASE IV. To add surd quantities together. RULE. When the surds are of the same kind, reduce them to their simplest forms as in the last case; then, if the surd part be the same in them all, annex it to the sum of the rational parts, and it will give the whole sum required. But if the quantities have different indices, or the surd part be not the same in each of them, they can only be added together by the signs + and EXAMPLES. 1. It is required to find the sum of 27 and 48. Here √27 = √(9 x 3) = 3√3 X And √ 48 = √(16 x 3) = 4√3 Whence 73 the sum. 2. It is required to find the sum of 3 500 and 108. Here 500 = (125 x 4) =5 3/ 4 And108= (27 x 4) = 3/4 Whence 8/4 the sum. 3. It is required to find the sum of 4 v 147 and 3 v75 Here 4 v 147 = 4√(49 x 3) = 28 √3 And 375 = 3√(25 x 3) = 15 v3 5. It is required to find the sum of √ 72 and v 128. 6. It is required to find the sum of 180 and v 405. Ans. 14 v (2). Ans. 15 (5). Ans. 93(5). 7. It is required to find the sum of 3340 and 135. 8. It is required to find the sum of 454 and 5128. Ans. 32 (2). 9. It is required to find the sum of 9243 and 10363 2 10. It is required to find the sum of 33 11. It is required to find the sum of 12/ 1 Ans. 191√(3). 1 32 Ans. 6 (2). 12. It is required to find the sum of a2b and √4bx*. CASE V. 1 Ans. (+) √b. To find the difference of surd quantities. RULE. When the surds are of the same kind, prepare the quantities as in the last rule; then the difference of the rational parts annexed to the common surd, will give the whole difference required. But if the quantities have different indices, or the surd part be not the same in each of them, they can only be subtracted by means of the sign -. 1. It is required to find the difference of ✓ 448 and 112. Here √ 448 = √(64 × 7) = 8√7 And √ 112 = √(16 × 7) = 4√7 X 3. It is required to find the difference of 5 Here 520 = 5 (4 × 5) = 10 √5 20 and 345. Whence 5 the difference. 4 It is required to find the difference of✓ and 4 1. It is required to find the difference of 250 and 2. It is required to find the difference of 3. It is required to find the difference of 4. It is required to find the difference of 2 5. It is required to find the difference of 3 6. It is required to find the difference of V 18. 40. Ans. 7 (2). 320 and Ans. 23(5). and √27 Ans.(15). and √8. Ans. √(2). and 372. Ans. 3(9). 2 and 3 9 32 Ans. (18). 7. It is required to find the difference of 80ax and √20α2x. Ans. (4a2-2ax) (5x). 8. It is required to find the difference of 8 ab and 2ab. Ans. (8a-2a2) / (b). Note. The two last answers may be written thus, (2ах - 4a2) √(5x), and or (4a2 2ax) √5x (8a 2a2) b. CASE VI. To multiply surd quantities together. RULE. When the surds are of the same kind, find the product of the rational parts, and the product of the surds, and the two joined together, with their common radical sign between them, will give the whole product required; which may be reduced to its most simple form by Case III. But if the surds are of different kinds, they must be reduced to a common index, and then multiplied together as usual. It is also to be observed, as before mentioned, that the pro |