13. a + b + c + 2 (a‡b¥ + a‡c‡ + b‡ct). p+! x 2 pq. 14. 4xy* – 12 g + 17 y* – 12 x 3 + 42. x3y x2y3 y 15. a2b-14 ab- 8a-1b + 4a-2b+ 8. Find the cube roots of 10 m a2n). 16. 24+ 9x+6 x 99 x2 - 42 x + 441 x 343. 17. x3y-1 + 3x1y− + 3x2y- + 1. 18. ab (1 + 3a−3b‡ + 3 a−‡b‡ + a−1b) (ab−1 – 3 a‡b−‡ + 3 a3b−3 + 1). CHAPTER IV. SURDS. 12. A surd quantity is one in which the root indicated cannot be denoted without the use of a fractional index. Thus, the following quantities are surds: Ja, Ja2 + x2, Na2 + b2 + c2, a + x x + y Since, from what has been explained in the last chapter, these quantities may be written thus a3, (a2 + x2)¿, (a2 + b2 + c2)*, (2 + 2)*, α (a + x)+ (x + y) + it follows that surds may be dealt with exactly as we deal with their equivalent expressions with fractional indices. It is evident that rational quantities may be put in the form of surds, and conversely, expressions which have the form of surds may sometimes be rational quantities. Thus, a2 = Y (a2)3 = Ya3; and a3 + 3ab + 3 ab2 + b3 = √(a + b)3 = a + b. 13. A mixed quantity may be expressed as a surd. Thus, 3/5 = 33. 3/5 = 33 × 5 = √135, 14. Conversely, a surd may be expressed as a mixed quantity, when the root of any factor can be obtained. Thus, = 18ab2 = √9a2b2 x 2 a √9 a2b2. √2a = 3 ab √2 a. And (a + b2)x+y3 = √(a2 + b2)x3y3 × xy2 15. Fractional surd expressions may be so expressed that the surd portion may be integral. The process is called rationalizing the denominator, and is worth special notice. It is much easier to find approximately the value of √21, and divide the result by 7, than to find the values of √3 and/7, and divide the former by the latter. Ex. 2. Reduce to its simplest form b xy с Ex. 3. Find the arithmetical value of 2 4 √3 The denominator is the difference of two quantities, one of hich is a quadratic surd. = = Now, we know that (2 √3) (2 + √3) 22 2a - (√3)3 = 4 - 3 1, and hence we see that by multiplying numerator and denominator by the sum of the quantities in the denominator we can obtain the denominator in a rational form. We shall now give an example when the surds are not quadratic. Ex 4. Rationalize the denominator of Since (x)12 α ys (y)12 is (Art. 29, page 175) divisible by xy, it follows that the rationalizing factor is their quotient, which is easily found. 16. Surds may be reduced to a common index. Ex. 1. Express /a and b as surds having a common index. = = 1 Since a am, and "/b b, it follows that, by reducing the fractional indices to a common denominator, the given surds become respectively amn, bmn, or "*/ɑ”, mn/¿m. Ex. 2. Reduce a3b and V3y2 to a common index. The least common denominator of the fractional indices of the given surds is 4 × 3 or 12. Hence we proceed as When the student has had a little practice, the first two steps of each of the operations may be omitted. 17. Addition and subtraction of similar surds. DEF. Similar surds are those which have the same irrational factors. Ex. 1. Find the sum of √12, 5 √27, We have √12 + 5 √√27 – 2 √75 = 2 √75. √22 x 3 + 5√32 x 32√5 x 3 = 2/3 + 5 x 33 - 2 x 5/3 = 2√3 + 15 √√/3 18. Multiplication and division of surds. a a + The following examples will best illustrate these opera tions: Ex. 2. Multiply a √b + c √d by a - √bd. Arranging as in the case of rational quantities, we have When the divisor is a compound quantity it will generally be the best to express the surds as quantities with fractional indices, and proceed as in ordinary division. 19.The square root of a rational quantity cannot be partly rational and partly irrational. If possible, let a = m + √o; that is, an irrational quantity is equal to a rational quantity, which is absurd. 20. To find the square root of a binomial, one of whose terms is a quadratic surd. then, squaring, a + √ b = x + y + 2√√xy,. ..(1); ..(2). Equating the rational and irrational parts (Art. 19), we |