Find the fourth roots of the following expressions. 29. 1+4x+6x2 + 4x3 + x^. 30. 16x1-96x3y +216x2y2 − 216xy3+81ya. 31. 1-4x+10x2 – 16x3+ 19xa— 16x2+10x6 — 4x2 + 28. 32. {x1 −2(a+b)x3 + (a2 + 4ab+b2)x2 – 2ab(a+b)x+a2b2}2. Find the eighth roots of the following expressions. 33. x8+8x7 + 28x6 + 56x5 +70xa +56x3 +28x2+8x + 1. 34. {x1-2x3y+ 3x2y2 — 2xy3 + ya}4. Find the square root of the following numbers. 35. 1156. 36. 2025. 37. 3721. 38. 5184. 39. 7569. 40. 9801. 41. 15129. 42. 103041. 43. 165649. 44. 3080.25. 45. 41-2164. 46. *835396. 47. 1522756. 48. 29376400. 49. 384524.01. 50. 4981 5364. 52. 24373969. 53. 55. 3-25513764. 57. 5687573056. 51. 64 128064. 144168049. 54. 254076·4836. Extract the square roots of each of the following num bers to five places of decimals. 59. 9. 60. 6.21. 61. 43. 63. 17. 64. 129. 65. 347 259. 62. 00852. 66. 14295 387. Find the cube roots of the following expressions. 67. 8.x3+36x2y +54xy2+81y3. 68. 1728x+1728x1y3 +576x2y° + 64y3. 69. x3-3x2(a + b) + 3x (a+b)2 — (a+b)3. 70. 26+3x2 + 6x2+7x3+6x2+3x+1. 71. x6-3αx2 + 5a3x3 – 3α3x — αo. 72. 8x6+48cx2 + 60c2x21 — 80c3x3- 90c2x2 + 108c5x-27c. 73. 1-9x+39x2— 99x3 + 156x1 — 144x3 +64x®. 74. 1-3x+6x2 – 10x3 + 12xa – 12x2 + 10x6 — 6x2 + 3x3 — x3. Find the sixth roots of the following expressions. 75. 1+12x+60x2 + 160x3 +240x2 + 192x2 + 64x6. 316. We have defined an index or exponent in Art. 16, and, according to that definition, an index has hitherto always been a positive whole number. We are now about to extend the definition of an index, by explaining the meaning of fractional indices and of negative indices. 317. If m and n are any positive whole numbers amxa"=am+n The truth of this statement has already been shewn in Art. 59, but it is convenient to repeat the demonstration here. ama×a×a× ....to m factors, by Art. 16, a" = a×a×a×......to n factors, by Art. 16; therefore a" × a" = a×a×a× ... ×a×a×a× ... to m+n factors, =am+", by Art. 16. In like manner, if p is also a positive whole number, am × a” × a2 = am+n× AP=am+n+p; and so on. 318. If m and n are positive whole numbers, and m greater than n, we have by Art. 317 am¬n × an = am−n+n=ɑm ; am therefore = am-n. a" This also has been already shewn; see Art. 72. 319. As fractional indices and negative indices have not yet been defined, we are at liberty to give what definitions we please to them; and it is found convenient to give such definitions to them as will make the important relation am xa" = am+n always true, whatever m and n may be. For example; required the meaning of a3. By supposition we are to have aa × a2 – a1= =α. Thus a must be such a number that if it be multiplied by itself the result is a; and the square root of a is by definition such a number; therefore a must be equivalent to the square root of a, that is, a3= √a. Again; required the meaning of a3. By supposition we are to have a×a×a3=a+} +} = a1= a. Hence, as before, a3 must be equivalent to the cube root of a, that is a3 = 3/a. Again; required the meaning of a. By supposition, a3× aa × aa × aa=a3 ; These examples would enable the student to understand what is meant by any fractional exponent; but we will give the definition in general symbols in the next two Articles. 1 320. Required the meaning of a where n is any positive whole number. therefore a" must be equivalent to the nth root of a, 321. Required the meaning of a" where m and n are any positive whole numbers. therefore a must be equivalent to the nth root of a, that is, n m a" = "/am. Hence a means the nth root of the mth power of a; that is, in a fractional index the numerator denotes a power and the denominator a root. 322. We have thus assigned a meaning to any positive index, whether whole or fractional; it remains to assign a meaning to negative indices. For example, required the meaning of a-2. We will now give the definition in general symbols. 323. Required the meaning of a ̄*; where n is any positive number whole or fractional. By supposition, whatever m may be, we are to have Now we may suppose m positive and greater than n, and then, by what has gone before, we have In order to express this in words we will define the word reciprocal. One quantity is said to be the reciprocal of another when the product of the two is equal to unity; thus, for example, a is the reciprocal of 1. Hence a" is the reciprocal of a"; or we may put this result symbolically in any of the following ways, 一路 324. It will follow from the meaning which has been given to a negative index that am÷a"-am-" when m is less than n, as well as when m is greater than n. For suppose m less than n; we have Suppose m=n; then a"a" is obviously = 1; and am-"=ao. The last symbol has not hitherto received a meaning, so that we are at liberty to give it the meaning which naturally presents itself; hence we may say that a = 1. |