8. So that the wines could be mixed in either of two ways--corresponding to the two different ways in which the gains and the losses could be made to counterbalance one another : gall. galls. (15; gain on i = 9 90 | 20 ; gain , 4. 8 20 = 160 245. Ans. 30 = 270 4, 32 128 6 @ 15 30; loss (32; loss = 6 4 در 97 » » gain 9 27 ,, 24 = 648 EXAMPLE IV.--There are three different kinds of rumworth 16s., 158., and 8s. a gallon, respectively; what quantity of each would be required to form 120 gallons of a mixture worth iis. a gallon ? We first find the proportions in which the three kinds should be mixed. On every gallon of them 16 rum, there would be a loss of 5 loss 3 In order, therefore, that the loss on the first rum may be counterbalanced by the gain on the third, the mixture should contain 5 gallons of the latter for every 3 of the former; and 5 that the loss on the second rum may be counterbalanced by the gain on the third, there should be 4 gallons of the 8s. for every 3 of the 158. kind : galls. r3 of the 16 rum= 3x58. Loss on s 15 - 3 x 48. Gain ,, 5+4 » 8 3x (5+4) s. gall. s. galls. 48 45 s. 3x55. Total loss=3x (5+4)s. 8= 72 15 » 165 IIS. 4= ܕܕ ܕܕ ور 9 II == Dividing 120 into three parts proportional to the numbers 3, 3, and 9, we thus find the required quantities to be 24 gallons of the 16s., 24 of the 15s., and 72 of the 8s. rum : 3+3+9=15; 24 : : : I 20 II = 1,320 Note.—The “proportions” found in the last example are by no means the only ones which would answer. Thus, if 7 gallons (say) of the 16s. and 13 gallons of the 15s. rum were taken, there would be a loss-on the former, of (7 X 5=) 358.; on the latter, of (13X4=) 528.; altogether, of (35+52=) 878. Dividing 878., therefore, by 38.—the gain on a gallon of the 8s. rum, we see that the necessary quantity of this rum would be (87+3=) 29 gallons: II = 539 So that if 120 were divided into parts proportional to 7, 13, and 29, those parts would fulfil the required conditions. POWERS AND ROOTS. 214. A number which is the product of two or more equal factors is said to be a POWER of one of those factors; and a number from whose repetition, as factor, a certain product can be obtained is called a ROOT of that product. We say that 49 is a power of 7, or that 7 is a root of 49 ; that 125 is a power of 5, or that 5 is a root of 125; that 1,296 5 is a power of 6, or that 6 is a root of 1,296; and so on : 49=7X7; 125=5X5X5; 1,296=6X6X6X6; &c. a 7=* 49 215. The number denoting how many times the root is contained, as factor, in the power is termed the INDEX. We say that 49 is the second power of 7, or that 7 is the second root of 49; that 125 is the third power of 5, or that 5 is the third root of 125; that 1,296 is the fourth power of 6, or that 6 is the fourth root of 1,296 ; and so on. Here the indices are 2, 3, 4, &c., and are written in either of the following two ways : 49=72 125=53 1,296=64 6= 1,296 &c. &c. &c. &c. It will be observed that 72, 53, and 64 are short expressions for 7 x7, 5X5 X5, and 6x6x6x6, respectively. It will also be observed that “ 49=72" and "7=/ 49" are two different statements of the one fact : 66 49=72" meaning that 49 is the second power of 7; and “7=349," that 7 is the second root of 49. Again, “125=53” and “5=3125" are two different statements of the one fact, and the remark is equally applicable to the expressions “1,296=64” and “6=V1,296": because, to say that 125 is the third power of 5 is exactly the same as 5 to say that 5 is the third root of 125; and to say that 1,296 is the fourth power of 6 is the same as to say that 6 is the fourth root of 1,296. NOTE.--In expressing the second root of a number, we usually omit the index (2). Thus, for the second root of 49 we write v 49, instead of 4 49. 216. The second power of a number is commonly spoken of as the SQUARE of the number, and the second root as the SQUARE root. This is explained by the fact that the area of a square is expressed by the second power of the number denoting the length of one of the sides. Thus, if the length of the side were 8 yards, the area of the square would be (82=8 x 8=) 64 square yards. 217. The third power of a number is generally spoken of as the CUBE of the number, and the third root as the CUBE root. This is accounted for by the fact that the solidity or capacity of a cube is expressed by the third power of the number denoting the length of one of the edges. Thus, if the edge were 4 inches in length, the cube would contain (43=4x 4x4=) 64 cubic inches. INVOLUTION. X 218. The finding of powers, when roots and indices are given, is called INVOLUTION. Every exercise in Involution, it is hardly necessary to observe, is worked by multiplication. Thus, to find the fifth power of 7, we simply set down 7, as factor, five times, and then multiply the factors together : 75 = 7*7*7 * 7*7= 16,807. Here 7, the root, would be said to have been raised (or “ involved") to the fifth power. 219. The product of two or more powers of the same number is, itself, a power of that number: a power whose index is the sum of the indices of the powers employed as factors. Thus, 87 x 8=(8 X 8 X 8 X 8 X 8 X 8 X 8) X(8 x 8 x 8 x8=811 —i.е., 87+4; 65 x 63 x6°= (6х6х6х6х6) х (6х6х6) x (6x6)=610-i.e., 65+3+2; &c. — c General formula: 2cm x x x xo=xn+b+c. 220. The index is doubled when a power is squared, trebled when a power is cubed, and so on. This follows from $ 219. The square of 52, for instance, is 52 x 52=54; the square of 53 is 53 x 58=5°; the square of 54 is 54 x 54558 ; &c. In like manner, the cube of 5 is 52 x 52 x 52=5® ; the cube of 53 is 53 x 53 x 53=5°; the cube of 54 is 54 x 54 x 5'=5; &c. General formula : nth power of xa =žaXn. 221. When a power of a number is divided by another power of the same number, the resulting quotient is a power of that number : a power whose index is what remains when the index of the power employed as divisor is subtracted from the index of the power employed as dividend. 811 87 x 84 This also follows from $219. For instance, 84 610 68 x 62 - i.e., 811-4; =6_i.e., 610–2; &c. 62 62 General formula : 84 =87 ar .29 537 222. When UNITY is expressed as a power of any number, the index is nought (o). In order to establish this fact, which follows from $221, we 53 have merely to divide any power by itself. Thus, s=58-3= 5°; 53 74 =74-t=7o; =x2-=xo; &c. But such fractions as 74 xa 74 &c. are each equal to 1—the division of a number by za? itself giving unity for quotient. Consequently, I is equal to 5°, to 7°, to ro, &c. 223. When a number is expressed as a power of itself, the index is 1. Thus, 5=51; 7=77; x=xl; &c. 224. When a number is resolvable into two or more factors, we can raise it to any particular power by raising each of the factors to that power, and multiplying those powers together. Thus, 77-=(11X7)P=I1X7X11X7=(I1X11) X(727) =11?x7; 1053=(7x5X3)3=7X5X3X7X5X3X7X5X3 =(7x7x7) x (5 x 5 x 5) x (3X3X3)=73 x 53 x 33; &c. 225. When a number ends with one or more ciphers, its square ends with twice as many. . Thus, 1302=(13 x 10)2= [$224] 132 x 102=169 X 100= 16,900 ; 3,400”= (34 x 100)2 = 34” x 1002=1,156 x 10,000 =11,560,000 ; 26,000?=(26 x 1,000) =26X 1,000o=676 x 1,000,000=676,000,000 ; &c. 226. When a number ends with one or more ciphers, its cube ends with three times as many. |