2. Reduce at, b, c), and df, to a common index. Ans. (a3), (012), (4), and (24)t. 3. Reduce va + 2, Va – X, and Va – 2, to a common index. Ans. Va + x), Wa — «)', and 'la? — x2)3. 4. Reduce V, 2, and 5V3, to a common index. Ans. 2007, V7, and 527. 5. Reduce ax, (bx), (cx)t, and (dx)ł, to a common index. Ans. (a??yle) T'?, (1620) ??, (0424) T't, and (2329) T?. 6. Reduce cxo, (dx)t, and (24), to a common index. . Ans. (022), (das)t, and (24)* % Reduce VT, V10, and 101, index. Ans. 49, V1000, and 18,1.. to common common 8. Reduce VI, V18, and 1331, to a index. Ans. VŽ, VĂ, and V11. II, FUNDAMENTAL OPERATIONS ON RADICALS. 1o. Addition of Radicals. 130. Radicals cannot be added unless they are similar. To determine when they are similar, we must reduce them to their simplest form; then, if their radical parts are the same, they will be similar, and if we regard the common radical part as a unit, we shall have the following rule for finding their sum: RULE. Reduce the radicals to their simplest forms; then, if they are similar, add the coefficients for a new coefficient, and write the sum before the common radical part. EXAMPLES. Find the sums of the following groups of radicals: 1. V18, V32, V50, and 772. Ans. 3V2 + 4V2 + 5V2 + 6V2 18V2. 2. 2V8, 3V50, and 6V18. Ans. 37V2. 3. V, 1ts, and VH. Ans. 3.1 15. 4. V, 107, and . Ans. 6. 5. XV 12a4x, 2a2v27x3, 3av 48a2x3, and 5a2x V3x. Ans. 25a x V3.2. 6. 54an+633, ař16an–376, and 2q4n+9. Ans. (3a2b + 262 + an+3) 2a". 7. 6V 4a, 22a, and V8a. Ans. 92a. 8. 2V3, 1V12, 4v27, and 2176 Ans. *V3. 9. 3620513, 92a365, and 8ay2d?65. Ans. 18ab2a-62. 131. We cannot subtract one radical from another unless the two are similar. In that case, we have the following RULE. Reduce the radicals to their simplest forms; then, if they are similar, subtract the coefficient of the subtrahend from that of the minuend, and write the remainder before the common radical part. EXAMPLES. 1. From 1320, subtract V80. Ans. 8V5 – 4V5 = 475. 2. From b27a6b, subtract 216a%b4. Ans. 3. From Va3 + 2ab + ab?, subtract Va3—2a2b + ab?. Ans. 20V a. 4. From fV+3V, subtract tha. Ans. 31. 5. From V 289a%b, subtract V144a. Ans. 5avi. 6. From 2v8a3+5 V 720%, subtract Yav 180 +V50ab%. Ans. (13a - 56) 2a. a + 2 1%. From (a − x) Va? – 22, subtract 8. From 81 + 192, subtract V512. Ans. 3 - 8. 30. Multiplication of Radicals. 132. Since two radicals can always be reduced to a common index, we may take año, and ca, to represent any two radicals whatever. The indicated product is, a VT xcņā. We may change the order of the factors without changing the value of the product; hence, may write the product under the form, aco x . But, from principle 1°, V x V = Vbd; hence, RULE. Reduce the radicals to a common index; then multiply the coefficients together for a new coefficient, and the quantities under the radical signs for a new quantity under the radical sign, leaving the index unchanged. % V8 x 5. Ans. V512 x 25 = 12800 = 2V200. By combining the above rule with that for the multiplication of polynomials, complicated radical expressions may be multiplied together. |