36. /a×a× Va. 37. (a2y) (ay). 38. (2x) x √(a3x). = 40. /a×a×a ̃ ̄ } ÷ a } . 42. (a2b1c5) 3⁄4/abc3). 44. √(ab1c1) × b ̄3× (c1a+) ̃§. Express the following quantities with radical signs and positive indices. 220. In the following examples the foregoing principles are applied, but the results of the operations of addition, etc., which have to be performed on the indices are given without any of the intermediate steps. Ex. 1. Multiply a3 +1 + a ̄3 by a3 − 1 + a ̄} 1 Ex. 2. Divide a+b+c-3abc by a + b3 +c3. α We perform the work by the synthetic method [Art. 83.]. - a3 (b3 + c3) + a3(b3 + c3 +2b3c3) − (b + c) -α Ex. 3. Find the square root of x3 – 4x+2x+4x − 4x3 + x3. 0. x3 − 4 x3 + 2 x3 + 4 x − 4x3 +x3 (xa − 2 xa + x03 12. x2n + xnyn + y2n by x-2n + x−¬ny-n + y-2n. 2 13. a+b+c c} _ b3c} _ c}a} – a}b3 by a3 + b3 + c3. 15. 81x3 – 27 x3y3 + 9x3y3 − 3x3y + y3 by 3x3 + y§. Divide: 4 23. a§ + a2b3 — a§b3 — ab + a1b3 + b3 by a3 + b3. 24. a -2+ at by a3- a. 25. (x − x−1) – 2 (x1 – x ̄3)+2(x3 − x ̄3) by x3 – x ̄‡. by x1y ̄3 +y3x ̄3. 27. Simplify (ab-35) × (a2b1c−1)} × (a−5bc2)§. 28. Simplify (a – 463) (a + 2 a3b3 + 4 b3) (a − 2 a3b3 +4b3). 31. Multiply 4x2 - 5x 4-7x-1+6x-2 by 3x-4 + 2x−1, and divide the product by 3 x − 10 + 10 x−1 — 4 x−2. (a + b) * + (a2 − b2)3 + (a − b)3 by (a + b)3 − (a — b)‡. 34. Write down the square roots of (i.) x+2x+1, (ii.) 4x3 – 4x3y3 + y3, ax3 — 2 a3x3 + a3x. (iii.) ax 35. Find the square root of 4 x2a-2-12 xα-1 + 25 — 24 x-1a + 16x-2α2. 36. Find the square root of 25 x2y-2 + y2x−2 — 20 xy-1 — 2 yx−1 + 9. 37. Find the square root of 1-375 +2a3x3 CHAPTER XXII. SURDS. COMPLEX QUANTITIES. 221. Definitions. A surd is a root of an arithmetical number which can only be found approximately. Thus 2 and 3/4 are surds. An algebraical expression such as a is also often called a surd, although a may have such a value that a is not in reality a surd. Surds are said to be of the same order when the same root is required to be taken. Thus √2 and 62 are called surds of the second order, or quadratic surds; also 3/4 and 5 are surds of the third order, or cubic surds; and 5 is a surd of the nth order. Two surds are said to be similar when they can be reduced so as to have the same irrational factors. Thus 2√2 and 5√2 are similar surds. The rules for operations with surds follow at once from the principles already established. NOTE. It should be remarked that when a root symbol is placed before an arithmetical number it denotes only the arithmetical root, but when a root symbol is placed before an algebraical expression it denotes any one of the roots. Thus a has two values, but √2 is only supposed to denote the arithmetical root, unless it is actually written ±√2. |