6. √98ax and √36x2 - 36 a2. Ans. 7a√2x+6√x2 — a2. 7. V98 a'x and √288 a*25. Ans. (7a+12a2x2)√2x. 15. 3b√2α3b2, 7√2a313, and 8a√2a2b3. Ans. 18 ab√2 a2b2. 16. √⁄8a3b+16 a1 and √b*+2ab3. Ans. (2a+b)√2a+b. 17. 6√4a2, 2√2a, and √8a3. Ans. 92a. 145. Radicals are subtracted like other algebraic quantities. Hence the following rule: If the radicals are similar, subtract the coefficient of the subtrahend from that of the minuend, and to the difference annex the common unit or radical. If the radicals are not similar, indicate the operation by the minus sign. Exercises. 1. What is the difference between 3 a√b and a√b? Here з a√b-a√b=2a√b. Ans. 2. From 9a√2762 subtract 6 a√27b2. Ans. 9 ab√3. 9 a√27 b2 = 27 ab√3, and 6a√27 b3 = 18 ab √3. NOTE. Two radicals which do not appear at first to be similar may become so by transformation, as in the preceding case. For example: 216 ab* -b 27 ab = √216 ab3 × b − b √ 27 a® × b NOTE. — Here 512 is not similar to the sum of 81 and 192; hence the operation of subtraction can only be indicated. Multiplication of Radicals. 146. Radicals are multiplied like other algebraic quantities. Hence we have the following rule : : Multiply the coefficients together, for a new coefficient. Multiply the radical parts together, for a new radical part. Then reduce the result to its simplest form. Exercises. 1. Multiply 3a√be by 2√ab. 3 a√bc × 2√ab = 3 a × 2 × √bc × √ab, NOTE. When the radicals have not the same index, they must be reduced to a common index before multiplying. For example: to multiply 2a√x by 3Vy, we must reduce them to the common index 6, making them 2a√x and 3Vy; and the product is 6a V3y2. 13. 3a8a and 264 a'c. Ans. 6 ab√32ac= 12ab2c. NOTE. By combining the foregoing method with the method for the multiplication of polynomials ( 43), complicated radical expressions may be multiplied together. 14. √x+2√x+4 and Vx+2√x. +2x+4x3 +2x+4x2+82 x2+4x2+8x3+8x+ NOTE. The two results are identical, but the second has been obtained by following the ordinary rule for exponents. Hence we conclude that the rule for multiplication is the same whether the exponents are entire or fractional. 15. (a*+a2x2+a*x+x3) × (a* − x3). Ans. a - x2. Ans. x2+px Ans. & 2 + √ a2 - c2. |