. 84 Given 20-32=5; and Given 28 + 9x=30, to find 2. 9 0 ....0 5(3.4259887573622 .... 4 30 (2.1808498 0 0 Div. 13 26 4 4 6.4..2.56 4.624 6.1.... 61 2.161 Div. 11.56 376 Div. 21.61 1.839 16 288488 1 1.819232 7.22...1444 87512 6.38....6104 19768 Div. 14.4244 73027625 Div. 22.7404 18610 4 14484375 64 1158 931 23.2624 227 7/21759...655 1172444 2|3.217 209 14.6484|2 841 128350 18 57 733 117250 Root, 18 x=2.1808498 Givenx+2+=100 find %. Given 2 12x = 12, to find x. 1 -12 -12(1.11574951 5.... 20 100 (4.2644 1.... 1 --11 676 Div. 21 - -1 16 16 869 1.32 .. 2.64 11.9 28 3.1.... .31 - -131 Div. 59.64 4.072 Div. 8.69 83369 04 3.788376 1 --47631 16.66...8196 283624 3.31 331 Div. 63.1396 256072 Div. 8.3369 -6195875 36 27552 1 55136 -5787588 25631 3.335.... 16675 408287 1921 Div. -8.287025 -330625 1282 25 13.792.. 5517 -78662 639 2342 --74390 64.078605 577 8.267983 276 56-4272 62 13 64.078881 1 83 4 31. Two travellers, A and B, set out together from the same place ; A travels 8 miles the first day, 12 the second, 16 the third, &c.; B goes 1 mile the first day, 4 the second, 9 the third, &c. ; how many days must they travel before B overtakes A. 7 days. -11 13.784... 16 3.3457.... of a. Of Irrational Quantities, of Surds. Irrational Quantities, or Surds, are those of which the values cannot be accurately expressed in numbers, or are such as have no exact root, and are usually expressed by means of the radical sign Ķ, or by fractional indices; in which latter case the numerator shows the power the quantity is to be raised to, and the denominator its root . Thus, 1/2, or 27, denotes the square root of 2; 2Na, or at the cube of the square of a, and am, is the mtb Va, a, root of the nth power CASE I. To reduce a rational quantity to the form of a Surd. RULE. Raise the quantity to a power corresponding with that denoted by the index of the surd to which it is to be reduced ; and over this new quantity place the radical sign, or proper index, and it will be of the form required. 1. Let 3 and 5 be reduced to the form of the square root. Here 3X3=3=9; whence 79. Ans, Here (5)=25; therefore 725, Ans. 2. Reduce 2.2 and 3x to the form of the cube root. Here (2x):8–876; whence w/8x®, or (82935 Here (32)_2728; therefore 9-27x, Ans. 5. Let —2a be reduced to the form of the fourth root. Here (2a)=16a"; therefore W16a", Ans. 6. Let a be reduced to the form of the fifth root, and wat wa NE, and to the form of the square root. 2a bna Here (a) =a"; therefore a", Ans. And (watozatb+2wab; :: W(a+b+2Nab). Again, 1 = 2a ; . Nda) 42:(ła). Also, wala pa i no 77 ° * A quantity of the kind here mentioned, as for instance 2, is called an irrational number, or a surd, because no number, either whole or fractional, can be found which, when multiplied by itself, will produce 2. But its approximate value may be determined to any degree of exactness by the common rule for exfracting the square root, being 1 and certain non-periodic decimals, which never terminate. Note. Any rational quantity may be reduced by the above rule to the form of the surd to which it is joined, and their product be then placed under the same index or radical sign. Thus 2x2=N4XNN (4x2)=18, a a 4a as 2a =w10=3 4a? . 9 8q 9 72a 72a 2a Here ラージラ× 108a Here 5/6=W25XW6= 150, Ans. Here $5a=N 25X50=N(), Ans. CASE II. To reduce quantities of different indices to others that shall have a given index. RULE.--Divide the indices of the proposed quantities by she given index, and the quotients will be the new indices for those quantities. Then, over the said quantities with their new indices, place the given index, and they will be the equivalent quantities required. 1. Reduce 31 , 1 or aand 6 si each to quantities that shall have the index . Here 1:5X=33, the first index ; And +=X=1=2, the 22 index. Whence (89) * and (2) ¢, or 276, and 46, are the quantities required. . Here 1 + $=x=3, 1st index, $=1= x=2, 2d index ; herce (59and (67!, or 125% and 366, are the answers sought: ) ': Here ixt=3, 1st index; =fx=4, 2d index. Therefore (a) and ()# are the answers. 3. Reduce 21 and 41, to quantities that shall have the common index $ Here + sixt=4, 1st index, -=X=2, 2d index; therefore (24# and (43_165 and 16% $ Ans. 166 and 165 4. Reduce c2 and a to quantities that shall have the common index 1. Here 4+1=x128, 1st index ; 1=1=X12, 2d. 子 Therefore (29) and (a?) are the quantities sought. Note. Surds may also be brought to a common index by reducing the indices of the quantities to a common denominator, and then involving each of them to the e, power denoted by its nume1. Reduce 34 and 4 or 4 and 54, or it and at, or at and ! 12 each to quantities having a common index. Here 31–38=(39)# =(39)&=(27), and 4 43=4&=(49)+=(1 42) Whenice (27;&and (16) +, Ans. Here 1 and 1 = t; and y's; hence (46) and (5)*, Ans. $ ) 59 =4 = and (6975, m n n m mn m n I Here f=fand f=8; therefore ()6 and (, Ans. I'm Here again j=ih and = id; therefore (a*) 2. Reduce a" and 3" to quantities that shall have a common index. 1 1 Here reduces to the common denominators and ; therefore (am) and (6")" Ans. RULE. Resolve the given number, or quantity, into two factors, one of which shall be the greatest power contained in it, and set the root of this power before the remaining part, with the proper radical sign between them.* 1. Let 48 and „108 each be reduced to its most simple form. Here 48=(16X3)=473, and 108=(27X4)=374. Note 1. When any number, or quantity, is prefixed to the surd, that quantity must be multiplied by the root of the factor above. mentioned, and the product then joined to the other part, as before. 1. Let 2X32 and 53/24 each be reduced to its most simple form. Here 2N32=2N (16X2)=812, & 5724—5 (8X3)=10%/3. Note 2. A fractional surd may also be reduced to a more convenient form, by multiplying both the numerator and denominator by such a number, or quantity, as will make the denominator a complete power of the kind required ; and then joining its root, with 1 put over it, as a numerator to the other part of the surd. 1 1. Let Vy, 125, 294, 56, V192, 7/80, 981, and TÄIN, each be reduced to its most simple form. Here „Ž=N=N(X14)=1N14, Ans. 3733 =//GX50)=50, Ans. „125 (25x5)=575, Ans. 294=N(49 X6)=75, Ans. 356=(8X7)=277, Ans. 192=64X3)=4/3, A. 7/80=7V (16x5)=2875, A. 9781= (27X3)=27/3, Ans. Here, reducing the radical, we have v = = $30; therefore TINTIX/30=zx/30, Ans. 10. Reduce 1698aRx, and (2_a*x*), each to the most simple form. Here Ni=No=112 ;; hence wi=112, Ans. Ņ98a+z=(49a2x2x)=7ax/2x, Ans. (2-a)={x*(x--a?)}=XN (a), Ans. *When the given surd contains no factor that is an exact power of the kind required, it is already in its most simple form. Thus, N15 cannot be reduced lower, because neither of its factors, 5 nor 3, is a square. 12 64 CASE IV. To add Surd Quantities together. RULE. When the surds are of the same kind, reduce them to their simplest forms, as in the last case; then, if the surd part be the same in each of them, annex it to the sum of the rational parts, and it will give the whole sum required. But if the quantities have different indices, or the surd part in each be not the same, they can only be added together by the signs + and 3N 1. It is required to find the sum of „27 and „48. 27=)=33 And 74&V (163= }=713 the sum. 2. It is required to find the sum of 500 and 108. A} .894 the sum. ()3 - İN Vio . =) Also V405=V81X5=975}=1575, Ans. =) 7. Required to find the sum of 3%40 and 135. First 30=3(8x5)=635 And 135=(27x5)=335) 5}995 =9/5, Ans. 8. Find the sum of 454 and 5128. Here 4/54347 (27X2)=122 And 5128=5764X2)=2092) ... 3272, Ans. 9. Find the sum of 9/243 and 10/363. 9 = 97) = And 10.7363–10/(121*3=11073}=191V/3, Ans. ) 10. Find the sum of 37 and 71. By first reducing the fractional surds, we have N}=N=16, and N=N=ion (9X6)=*N6; Hence 33 X 116=16 = } And I=7X16=476) VE}=w6, Ans. iI 1 |