Q. In multiplying 3 by 3, making 9, and the 9 also by 3, making 27, we use, the 3 three times; what, then, is the 27 called? A. The third power, or cube of 3. Q. What is the third power of 2? 3? 4? 1 Q. What is the figure, or number, called, which denotes the power, as 3d power, 2d power, &c.? A. The index, or exponent. Q. When it is required, for instance, to find the third power of 3, what is the index, and what is the power? A. 3 is the index, 27 the power. Q. This index is sometimes written over the number to be multiplied, thus, 32; what, then, is the power denoted by 24? A. 2× 2 × 2 × 2 = 16. Q. When a figure has a small one at the right of it, thus, 65, what does it mean? A. The 5th power of 6, or that 6 must be raised to the 5th power. 1. How much is 122, or the square of 12? A. 144. 2. How much is 42, or the square of 4? A. 16. 3. How much is 102, or the square of 10? A. 100. 8. What is the cube of??? A. 1. 24. 4. 9. What is the square of ,5? 1,2? A.,25. 1,44. 11. Involve to the 2d power; 12. Involve to the 24 power. 13. Involve to the 2d power. 2 to the 2d power. A... A. A.. 14 What is, or the square of 22 15. What is the value of ? Exercises for the Slate. 1. What is the square of 900 ? A. 819000. 2. What is the cube of 211? A. 9393931. 3. What is the biquadrate, or 4th power, of 80? A. 40960000. 4. What is the sursolid, or the 5th power, of 7? A. 16807. EVOLUTION. 1 LXXXV. Q. What number, multiplied into itself, will make 162 that is, what is the first power or root of the square number 16? A. 4. Q. Why? A. Because 4 × 4—16. Q. What number multiplied into itself three times, will make 27? that is, what is the 1st power or root of the cubic number 27 ? A. 3. Q. Why? A. Because 3 x3x3=27. Q. What, then, is the method of finding the first powers or roots of 21, 3d, &c., powers called? A. Evolution, or the Extraction of Roots. Q. In Involution we were required, with the first power or root being given, to find higher powers, as 2d, 3d, &c., powers; but now it seems, that, with the 2d, 3d, &c., powers being given, we are required to find the 1st power or root again; how, then, does Evolution differ from Involution? A. It is exactly the opposite of Involution. Q. How, then, may Evolution be defined? A. It is the method of finding the root of any number. 1. What is the square root of 144? A. 12. Q. Why? A. Because 12 × 12=144. 2. What is the cube root of 27? A. 3. Q. Why? A. Because 3x3x3=27. 3. What is the biquadrate root of 81? A. 3. Q. Why? A. Because 3×3×3×3=81. We have seen, that any number may be raised to a perfect power by Involution; but there are many numbers of which precise roots cannot be obtained; 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 decimals, however, we can come nearer and nearer, that is, approximate 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 this character, 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 extracted; 4/64 the 4th root of 64. When we wish to express the power of several numbers that are connected ogether by these signs, +, -, &c., a vinculum or parenthesis is used, drawn from the top of the sign of the root, and extending to all the parts of it; thus, the cube root of 30-3 is expressed thus, 3/30-3, &c. EXTRACTION OF THE SQUARE ROOT. 1 LXXXVI. Q. We have seen (T LXXXV.) that the root of any number is its 1st power; also that a square is the 2d power: what, then, is to be done, in order to find the 1st power; that is, to extract the square root of any number? A. It is only to find that number, which, being multiplied into itself, will produce the given number. Q. We have seen (T LXXIX.) 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. Why? A. Because 3 feet X 3 feet 9 square feet. Q. What, then, is the difference in contents between a square figure whose sides are each 9 feet in length, and one which contains only 9 square feet? A. 9 X 981-972. 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 a figure, which contains 144 square feet? A. 12 square feet. Q. Why? A. Because 12 X 12-144. Q. How, then, may we know if the root or answer be right? A. By multiplying the root into itself; if it produces the given number, it is right. Q. If a square garden contains 16 square rods, how many rods does it measure on each side? and why? A. 4 rods. Because 4 rods X 4 rods 1. What is the square root of 64? and why? 7. What is the square root of ,25? A.,5. 8. What is the square root of 1,44? A. 1,2. =16 square rods. 9. What is the value of 25? or, what is the square root of 25? 10. What is the value of,/,4? A.,2. 11. What is the square root of? A. . 12. What is the value of? A.. 13. What is the square root of of? A. f. 14. What is the square root of 6? √6=√25=3=24, Ans. 15. What is the value of of? A. z. 16. What is the square root of 30? 17. What is the difference between the square root of 4 and the square of 4? or, which is the same thing, what is the difference between 4 and 42? 18. What is the difference between 19. What is the difference between 20. What is the difference between 21. There is a square room, which is calculated to accommodate 100 scholars; how many can sit on one side? 22. If 400 boys, having collected together to perform some military evolutions, should wish to march through the town in a solid phalanx, or square body, of how many must the first tank 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 size; I demand how many are contained in one of its sides? A. 40. 25. A man is desirous of making his kitchen garden, containing 2 acres, or 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. Exercises for the Slate. 1. If a square field contains 6400 square rods, how many rods in length does it measure on each side? A. 80 rods. 2. How many trees in each row of a square orchard, which contains 2500 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 runk and file, to form them in a complete square? A. 100 men. 4. What is the square root of 2500? A. 50. 5. What is the 1st power of 10000002? A. 1000. 6. What is the value of 360000? A. 600. 7. What is the difference between the square root of 36 and the square of 36? A. 1290. 8. What is the difference between 9. What is the difference between 10. What is the difference between 4900 and 49002? A. 24009930. 12. What is the amount of 4 and 9? A. 5. 13. What is the sum of 4 and 92? A. 83. 14. What is the amount of 301 and 272? A. 29. 15. What is the length of one side of a square garden, which contains 1296 square rods? in other words, what is the square root of 1296? In this example, we have a little difficulty in ascertaining the root. This, perhaps, may be obviated by examining the figure on the following page, (which is in the form of the garden, and supposed to contain 1296 square rods,) and curefully noting down the operation as we proceed. 900, sq. rods in A. 180 30 rods. 6 rods. 6 rods. 30 rods. form of a quotient in division; 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 rods, viz. 1296. This figure is divided into parts, called A, B, C, and D. It will be perceived, that the 900 square rods, which we deducted, are found by multiplying the length of A, being 30 rods, by the breadth, being also 30 rods, that is, 302=900. To obtain the square rods in B, C, and D, the remaining parts of the figure, we may multiply the length of each by the breadth of each, thus; 30 x 6180, 6 x 636, and 30 X 6 180; then 180 +36 +180=396 square rods; or, add the length of B, that is, 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 length 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 × 6=396 square rods, the same as before. We do the same in the operation; that is, we first double 30 in the quotient, and add the 6 rods to the sum, making 66 for a divisor; next, multiply 66, the divisor, by 6 rods, the width, making 396; then, taking 396 from 396 leaves 0. The pupil will perceive, the only difference between the 1st and 2d operation (which see) is, that in the 2d we neglect writing the ciphers at the right of the numbers, and use only the significant figures. Thus, for 30+6, we 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+6, we write 66. Omitting the ciphers 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 determine 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 contains but twice as many, or at least but one figure less than twice as many, figures as are in the root. Thus, the square of the root 30 is 900; now, in 900 there are but three figures, and in 30, two figures; that is, the square of 30 contains but one figure more than 30. We will take 99, whose square is 9801, in which there are four figures, and in its root, 99, but two; that is, there are exactly twice as many figures in the square 9801 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 the figures in the given number into periods of two figures each, commencing vith 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 begin at the units' place, and point off the figures towards the right, in the same manner as we point off whole numbers towards the left. By each of the preceding operations, then, we find that the root of 1296 is 36, or, in other words, the length of each side of the garden is 36 rods. |