Q. In multiplying 3 by 3, making 9, and the 9 also by 3, making 27, we do the 3 three times what, then, is the 21 alie!? A. The third power, or cube of 3. 4. What is the thiru power of 2? 3? 4? A. 8. 27. 04. Q. What is the figure, or number, called, which denotes the power, thus, power, 21 power, &c.? A. The inder, or exponent. ' Q. When i: is required, for instance, to find the third power of 3, what to che index, aod what is the power? A. 3 is the index, 27 the power. Q. This index is sometimes writion over the number to be multiplied thor 38 ; what, then, is ihe power denoted by 24? A. 2 X 2 X 2 X 2=10. 4. The 5th power of 6, or thai 6 must be raised to the 5th power. 1. Ilow much is 122, or the square of 12? A. 144. 2. How much is 4*, or the square of 4. A. 16. 3. Ilow much is 10%, or the square of 10? A. 100. 4. How much is 49, or the cube of 4? A. 64. 5. How much is 14, or the 4th power of I? A. I. 6. What is the biquadrate, or 4th power, of 3? A. 81. 7. What is the square of 1? ? A. . 8. What is the cube of ?? ? ? 4. . eks. 9. What is the square of j5? 1,2? A. ,25. 1,44. 10. Involve 2 to the 20 power; 2 tu the 3d power, A. 4, 8. 11. Involva fro the 241 power; it to the 24 power. A. o. T 12. Involve to the 21 power. A. t = 1. 13. Involve ff to the 21 power. A. $. 14. What is it, or the acuare of} ? A. 15. What is the valuo of ? A. ds. - 16. What is the value of '? A. g'T: Exercises for the Slate. 1. What is the square of 900 : A. 810000. 2. What is the cribe of 211: A. 9393931. 3. What is the biquinirate, or 4th power, of 80 ? A. 40960000. 4. What is the-sursolid, or the 5th power, of 7? A. 16807. 5. Involve ti, f, g, each w stie 34 power. A. 1731, 418, 6. What is the square or 54 ? 7. What is the square of 104? A. 272. 8. What is the value of 86 ? A. 32708. 9. What is the value of 104? A. 10000. 19. What is the value of ? A. 1206. U. What is thy suba of 5! A 519) EVOLUTION. on 34, 8CC.. ther., down the opp no defineile root of I LXXXV. Q. What number, multiplied into itsolf, will make 10+ that is, what is the first powor ur root of the square number 16? A. 4. . A. Because 4 X 4 = 16. * Q. What number multiplied into itself three timos, will make 27? that ., wbat is the 1st power or root of the cubic number 27? A. 3. Q. What, then, is the method of finding tho first powers of roots of 2d, 34, Rc., powers called? 1. Evolution, or the Extraction of Roots. Q. In Involution we were required, with :he first power or root being given, to find higher powers, as 2d, 311, kcc., powers; but now it seems, that, with tho 2d, 3d, &c., powers being given, we are required to find tho Ist power or roof again; how, ther, does Evolution differ from Involution ? A. It is exactly the opposite of Involution. Q: llow, then, may Evolution be defined ? A. It is the method of finding the root of ar y number. 1. What is the square root of 144 A. 12. Q. Why? A. Because 12 x 12=144. 2. What is the cube root of 27 ? A. 3. * Q. Why? . A. Because 3 X3 X3=27. * 3 What is the biquadrate root of 81? A. 3. Q. Why? A. Because 3 X3X3 X3=81. We have seen, that any number may be raised wa perfect power by Into fution ; but there are many numbers of which pracise roots cannot be obiained; as, for instance, the square root of 3 cannot be exactly determined, there being no number, which, by being multiplied into itself, will make 3. By the aid of deciinals, however, we can come nearer and nearer, that is, approximato towards the root, to any assigned degree of exactness. Those numbers, whose roots cannot exactly be determined, are called SURD Roots, and those, whose roots can exactly be determined, are called RATIONAL Roots. To show that the square root of a number is to be extracted, we prefix thin character, v. Other roots are denoted by the same character with the index of the required root placed before it. Thus, „9 signifies that the square root of 9 is to be extracted; 3. 27 signifies that the cube root of 27 is to be exuncted ; 64=the 4th root of 6i. When we wish to express the power of several numbers that are connocted kogether by these signs, +, -,&c., a vinculum or parenthesis is used, draw froin the top of the sign of the root, and extending to all the pa.ts of it; tbwe (be cube root of 30 -- 3 is expressed chun, /30 -3, de EXTRACTION OF THE SQUARE ROOT. | LXXXVI. Q. We have seen (L.XXXV.) that the root of any nomner is iu lat power; alou that a square is the 20 power: what, then, is lo be done, ip order to find the lat power; that is, to extract the square root of any number? A. It is only to find that nuniber, which, being multiplied into itself, will produce the given number. Inom Q. We have seen (1 1.XXIX.) that the process of finding the contents of a square consists in multiplying the length of one side into itself; when, then, the contents of a square are given, how can we find the length of each side! or, to illustrate it by an example, If the contents of a square figure be 9 feet, what must be the length of each side A. 3 feet. Q. What, then, is the difference in contents between a square figure whone sides are each 9 feet in length, and one which contains only 9 square feet? A. 9 XI=81-9=72. Q. What is the difference in contents between a square figure containing 3 square feet, and one whose sides are each 3 feet in length ? A. 6 square feet. Q. What is the square root of 144? or what is the length of each side of figure, which contains 144 square feet? A. 12 square feet. A. By multiplying the root into itself; if it produces the given number, it is right. Q. If a square garden centains 16 square rods, how many rosla does it measure on each side and why? A. 4 rods. Because 4 rods x 4 rods = 16 square rods. 1. What is the square foot of 64 ? and why? 2. What is the square tuut of 100? and why? 3. What is the square root of 49? and why? 4. Extract the square root of 141. * 5. Extract the square root of 36. 6. What is the square root of 3600? 7. What is the square root of ,25 A. :: 8. What is the square root of 1,44? A. 1.8.! . !! ; 9. What is the value of 25 ? 'or, what is the square root of 10. What is the value of 1,4? A. ,2. 11. What is the square root of 1?, A. 13. What is the value of w ? A.. 13. What is the square root of off? A. 14. What is the squase root of 1? vol=v4=f=2, Sea 15. What is the value of 16. What is the square root 17. What is the difference between the square root of 4 and the sqaure of <, which is the name thing, what is the difference between 4 and ? JA= 2 and d = 16; then, 16-9 = 14 A 18. What is the difference between 9 and 9?? 21. There is a square room, which is calculated to accommodate 100 schal eng; how many can sit on one side ? 2. If 400 boys, having collected together to perform some military evoke tions, should wish to march through the town in a solid phalanx, or square body, of how many must the first rank consist? 23. A general has 400 men; how many must he place in rank and file to form them into a square ? 24. A certain square pavement contains 1600 square stones, all of the same aize : Idemand how many are contained in one of its sides ? A. 40. 25. A man is desirous of making bis kitchen garden, containing 2% acres, 400 rods, a complete square; what will be the length of one side 26. A square lot of land is to contain 22% acres, or 3600 rods of ground; but, for the sake of fruit, there is to be a smaller square within the larger, which is to contain 225 rods : what is the length of each side of both squares ? A. 60 rods the outer, 15 rods the inner by the Exercises for the Slate. 1. If a square field contains 6400 square rods, how many rods in length dom it measure on each side? A. 80 rods. 2. How many trees in each row of a square orchard, which contains 25% trees? A. 50 trees. 3. A general has a brigade consisting of 10 regiments, each regiment of 10 companies, and each company of 100 men : how many must be placed in rank and file, to forin them in a compleie square ? A. 100 men 4. What is the square root of 2500? A. 50 7. What is the difference between the square root of 36 and the square 36? A. 1290. 8. What is the difference between 4900 and 4900 ? A. 24009930 11. What is the difference between 1 and 1962? A. H 15. What is the length of one side of a square garden, which In this example, we have a little difficulty in ascertaining the root. This perhaps, may be ohviated by examining the figure on the following page, (which is in the forin of tho garden, and supposed to contain 1206 squaro rodi,) and tarofully noting down the operation us we proceed OPERATIONS. ' In this example, we know that Tatt . 22. the root, or the length of one side Square Rods. Square Roda. of the garden, must bo greater 30 ) 1296( 30; 3) 1996 ( 36 than 30, for 30% = 900, and loud 900 than 40, for 40 = 1000, which greater than 1296 ; thoroforo, wo Tako M), the loss, and, for conta 306 ninnce' sako, writo it at the lake of 1296, as a kind of divisor, lila vine u the right of 1208, ha FIG. form of a quotient in division; 30 rods. 6 rods. (See Operation Ist.); then, subtracting the square of 30, = 900 8q. rods, from 1296 sq. rods, leaves 396 sq. rods. The pupil will bear in mind, that the Fig. on the left is in the form of the garden, and contains the same number of square ruds, viz. 1296. This figure is divided into parts, called A, B, C, and D. It will be perceived, that the 900 Rods. squarc rods, which we deducted, 30, length of A. are found by multipl;ing the 30, breadth of A. length of A, being 30 roils, by the breadth, being aišu 30 rods, then 900, aq. rods in A. is, 30% = 900. To obtain the square rods in B, C, and p, the remaining parts of the figure, we may multiply 30 ruda. 6 rods the length of each by the breadth of each, thus; 30 x 6= 180, 6 x 6 = 36, and 30 x 6= 180 ; then 180 + 36 + 180 = 396 square rods; or, add the length of B, that 19, 30, to the length of D, which is also 30, making 60; or, which is the same thing, we may double 30, making 60; to this add the longth of C, 6 rods, and the sum is 66. Now, to obtain the square rods in the whole length of B, C, and D, we multiply their length, 6 rods, by the breadth of each side, thus, 66 X 6= 396 square rode, the same as before, Wo do the same in the operation ; that is, we first double 30 in the quo Hiont, and add the 6 rods to the sum, making 66 for a divisor; next, muliiply 86, the divisor, by 6 rods, the width, making 396; then, taking 396 from 396 Daves 0. The pupil will perceive, the only difference between the lit and 22 operadion (which nee) is, that in the 2d we neglect writing the ciphers at the right of the numbers, and use only the sigincant figures. Thus, for 30 + 6, wo Write 3 (tens) and 6 (units), which, joined together, make 36; for 900, we write 9 (hundreds). This is obvious from the fact, that the 9 retains its place under the 2 (hundreds). Instead of 60 + 0, we writA 66, Omitting the oiphers in this manner cannot reasonably make any difference, and, in fact, it does not, for the result is the same in both. By neglecting the ciphers, we may, perhaps, be at a loss, sometimes, to de Carmine where we must place the square number. In the last example, we knew where the square of the root 3 (tens) = 9 (hundreds) should be placed, for the ciphers, at the right, indicate it ; but had these ciphers been dropped, we should, doubtless, have hesitated in assigning the 9 its proper place. This difficulty will be obviated by observing what follows. The square of any number never contatrs but twice as many, or at least but ono figure less than twice as inany, figures as are in the root. 'I hus, the aquare of the root 30 is 900 ; now, in 900 there are but three figues, and in 30, (wo figures; that is, the square of 30 contains but one figure more than 30, We will take 99, whose square 18 9001, in which there are four figures, and in ita root, 99, but tw; thut is, there are exactly twice as many figures in the squan 0801 as are in its root, 99. This will be equally true of any numbers whatever. Hence, to know where to place the several square numbers, we may point off the figures in the given oumber into periods of two figures each, commencing with the units, and proceeding towards the left. And, since the value of both whole numbers and decimals is determined by their distance from the units' place, consequently, when there are decimals in the given number, we may bogin'at the units' place, and point of the figures towards the right, in the saws Tanner as we point off whole numbers towards the left. By cach of the precuding operations, then, we find that the root of 1295 is Rot, in other words, the length of each side of the garden u 30 rolle. |