required; and then joining its root, with 1 put over it, as a numerator, to the other part of the surd. (s) EXAMPLES. 1. Let be reduced to its most simple form. 7 2. Let 33 be reduced to its most simple form. 5 3. Let 125 be reduced to its most simple form. 4. Let 294 be reduced to its most simple form. 5. Let 3/56 be reduced to its most simple form. 6 Let/192 be reduced to its most simple form. 7. Let 7/80 be reduced to its most simple form. 8. Let 981 be reduced to its most simple form. 3 9. Let N be reduced to its most simple form. 121 5 6 (8) The utility of reducing surds to their most simple forms, in order to have the answer in decimals, will be readily perceived from considering 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 pu pose) and then divide it by 7; whereas, otherwise, we must have first divided the numerator by the denominator, and then have found the rout of the quotient, for the surd part; or else have deterrained the root both of the numerator and denominator, and then divided the 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. 3 16 be reduced to its most simple form. 98a2x be reduced to its most simple form. x3-a2x2 be reduced to its most simple form 12. Let 3 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. Heré √27=√9X8 =3√3 Whence 7/3 the sum 2. It is required to find the sum of 3/500 and 2/108. Here /500=3/125X4=53/4 3/1083/27X4—33/4 Whence 83/4 the sum. 3. It is required to find the sum of 4/147 and 4. 'It is required to find the sum of 3/ 10 3 of 3 and 2√ Here 3/3= ✓10 25 5 :1 10 EXAMPLES FOR PRACTICE. 5. It is required to find the sum of 72 and 128. 6. It is required to find the sum of 180 and 405. 7. It is required to find the sum of 33/40 and 3/135. 8. It is required to find the sum of 43/54 and 53/128. 9. It is required to find the sum of 9/243 and 10/363. 2 27 10. It is required to find the sum of 3 3 of 3 and 7✓ 50 1 1 11. It is required to find the sum of 123/- and 33/ 4 32 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 And 44864x7=8/7 112/16X7=4√7 Whence 4/7 the difference. 2. It is required to find the difference of 3/192 and 3/24. Here /192/64X3=43/3 And 3/24 3/8×3 =23/3 Whence 23/3 the difference. 3. It is required to find the difference of 5/20 and 3/45. EXAMPLES FOR PRACTICE. 1. It is required to find the difference of 2/50 and 18. 2. It is required to find the difference of /320 and /40. 3. It is required to find the difference of✔ 3 and √ 5 9 and 8. 4. It is required to find the difference of 2 5. It is required to find the difference of 3 3/4 and 3/72. 2 3 6. It is required to find the difference of and 9 32 7. It is required to find the difference of80a4x and ✔20a3 x3. 8. It is required to find the difference of 8a3b and 23/ab. 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 product of different powers, or roots, of the same quantity, is found by adding their indices. |