4. Find the product of 5 + 2 x2a +3x-2a and 4xa - 3x 5. Divide 21x + x3 + x + 1 by 6. Divide 15 a 3 a3 - 2 a ̄ +8a-1 by 5a3 +4. 7. Divide 16 a¬3 + 6 a ̄2 + 5 a-1- 6 by 2 a-1 1. 8. Divide 2 1 56-66-46-46-5 by ba – 26. b 9. Divide 21 a3≈ + 20 – 27 a≈ - 26 a2x by 3 a 5. aš _ 3 4 a ̄ + aa+4a ̄*. 17. Multiply 2 Va—a— by 2 a -3/1-a. а 18. Divide V2+2x3- 16x-32 by x2+4x- + 2 х 19. Divide 1-√a +2a2 by 1 – a2. a-1 3 +36. — 23. 81 Vy 1-3 3/4-63/y. y 226. The following examples will illustrate the formulæ of earlier chapters when applied to expressions involving fractional and negative indices. The product = {2x2 - (x − 3)}{2x2 +(x − 3)} Ex. 3. The square of 3x2 =9x+4+x ̄1 −2·3x2. 2 -2.3x2. x ̄ ̄ − 9 x + 4 + x−1 − 12 x13 − 6 + 4x = + 2.2.x ̄ CHAPTER XXIII. SURDS (RADICALS). 227. A surd is an indicated root which cannot be exactly obtained. Thus √2, 3/5, a3, √a2 + b2 are surds. By reference to the preceding chapter it will be seen that these are only cases of fractional indices; for the above quantities might be written. 24, 51, at, (a2+by+. Since surds may always be expressed as quantities with fractional indices they are subject to the same laws of combination as other algebraic symbols. 228. A surd is sometimes called an irrational quantity; and quantities which are not surds are, for the sake of distinction, termed rational quantities. 229. Surds are sometimes spoken of as radicals. This term is also applied to quantities such as √a2, √9, No27, etc., which are, however, rational quantities in surd form. 230. The order of a surd is indicated by the root symbol, or surd index. Thus /, /a are respectively surds of the third and nth orders. The surds of the most frequent occurrence are those of the second order; they are sometimes called quadratic surds. Thus √3, √α, √x+y are quadratic surds. 231. A mixed surd is one containing a factor whose root can be extracted. This factor can evidently be removed and its root placed before the radical as a coefficient. It is called the rational factor, and the factor whose root cannot be extracted is called the irrational factor. 232. When the coefficient of the surd is unity, it is said to be entire. 233. When the irrational factor is integral, and all rational factors have been removed, the surd is in its simplest form. 234. When surds of the same order contain the same irrational factor, they are said to be similar or like. 5√3, 2√3, §√3 are like surds. Thus But 3√2 and 2√3 are unlike surds. 235. In the case of numerical surds such as √2, 3/5, although the exact value can never be found, it can be determined to any degree of accuracy by carrying the process of evolution far enough. that is 5 lies between 2.23606 and 2.23607; and therefore the error in using either of these quantities instead of √5 is less than .00001. By taking the root to a greater number of decimal places we can approximate still nearer to the true value. It thus appears that it will never be absolutely necessary to introduce surds into numerical work, which can always be carried on to a certain degree of accuracy; but we shall in the present chapter prove laws for combination of surd quantities which will enable us to work with symbols such as √/2, 3/5, a, ... with absolute accuracy so long as the symbols are kept in their surd form. Moreover it will be found that even where approximate numerical results are required, the work is considerably simplified and shortened by operating with surd symbols, and afterwards substituting numerical values, if necessary. |