253, 254, 255. INYOLUTION 161 11. How would you divide a | a common denominator ?(239)-10 whole nuinber by a fraction ?(225) | the least common denominator? - a fraction by a fraction ? 16. How are fractions of a bigher 12. How may you enlarge the denomination changed to a lower terms of a fraction ? (229) How di- | denomination ? (243)-into integer, minish them? of a lower?-a lower denomination 13. How would you find the to a higher-into integers of a greatest common divisor of two | higher . " numbers ? How reduce a fraction | 17. Is any preparation necessary to its lowest terms? i in order to add fractions ? (243) * 14. How would you find a com- 1 why must they have the sanie demon multiple of iwo numbers ?(236) 1 numinator ? low are they allded : the leasi common multiple ?" How is subtraction of fractions per13. How are fractions brought to formed? How the rule of three? SECTION VUI. POWERS AND ROOTS.. ANALYSIS. 254. Again, if the square, B, be multiplied by its root, A, ine product, (9*3=)27 feet, is the volume, or content, of the cube, A CE, p. 51, (01.) which measures 3 feet on every side. Hence if a line or a number be multiplied twice into itself, it is said to be cubed, or because it is employed three times as a factor, (33*3=27) it is said to be raised 10 ine third power, and the line or number which shows the dimensions of the cube, is called its cube root. Thus the cube root of ACE=27, is A03. 255. Again, if the cube, D, be multiplied by its TOOL, A, the product, (27X33) 81 feet, is the content of a parallelopipedon, ACE, whose length is 9. feet, and other dimensions, 3 feet each way, equal to 3 cubes, A CE, placed end to end. Hence if a given number be multiplieil 3 times into itself, or i/ employed four times as a factor, (3*3*3*3=81) it is raised to thc fourth power, or biquadrate, of which the girea nun. der is called the fourth root. 256. Again, if the biquadrale, D, be multiplied by its root, A, the pro- . duct, (81x3=) 243, is the content of a plank, equal to 9 cubes, ACE, laid down in a square form, and called ihe sursolid, or fifth power, of which A is the fifth root. 257. Again, if the sursolid, or fifth power, be multiplied by its root A, the product, (2433=) 729. is the content of a cube equal to 27 cubes, A CE, and is called a squared cube, or sixth power, of which A is the sixth root, 258, From what precedes it appears that the form of a root, or first power is a line, the second power a square, the third power a cube, the fourth power a parallelopipedon, the fifth power a plank, or square solid. and the sixth power a cube, and proceeding to the higher powers, it will be seen tbat the forms of the 3d, 4th and 5th powers are continually repeated; that is, the 31, 6th, 9th, &c. powers will be cubes, the 4th, 7th, 10th, &c. parallelopipedons, and the 5th, 8th, 11th, &c. planks. The rais. ing of power of numbers is called INVOLUTION 259. The number which denotes the power to which apother is to be raised, is called the index, or exponent of the power. To denote the second power of 3 we should write 32, to denote the 3d power of 5 we should write 53, and others in like manner, and to raise the number to the power required, multiply it into itself continually as many times, less one, as are deuoted by the index of the power, thus: 3=3 =3 first power of 3, the root. 32=3X3 =9 second power, or square of 3. 33=3X3X3 = 27 third power, or cube of 3. 34=3X3X3X3=81 fourth power, or biquadrate of 3. QUESTIONS FOR PRACTICE. 1. What is the fifth pow. 2. What is the second er of 6 ? power of 45 ? Ans. 2025. 3. What is the square 1 of 0.25 feet? (121) 36 2d power. Ans. 0.0625 ft. 4. What is the square of 216 3d power. inch ? Ans. fin. 5. What is the cube of 1296 4th power. | 1į, or 1.5 ? | Ans. 27-=3 3, or 3.375. 16. How much is 44 ? 62? - Ang. 7776 5th power. 1 836 76? 114 ? 1010? 280_264. EVOLUTION. · 163 260. The powers of the nine digits, from the first to the sixth inclusive, are exhibited in the following TABLE. Roots, or 1st powers 11 21 Squares, or 20 powers 1 4 9 16 253614964 -81) Cubes, or 31 powers 11! 8! 271 641 125 216 343) 5121729 Biquadrates, or 4th p.li/16) 811 2561 625) 12961 2401 40966561 Sursolids, or 5th pow.17132 243 1024, 3125 7776 168071 32768! 59049 Square cubes, or 6 p. 1764 729 4096|15625 46656117649 262144 531441 2. Evolution. ANALYSIS. 261. The method of ascertaining, or extracting the roots of cumbers, or powers, is called Evolution. The root of a number, or power, is a number, which multiplied by itself continually, a certain number of times, vill produce chat power, and is named from the denomination of the porer, as the square root, cube root, or 2d root, 3d root, &c. Thus 27 is the cube or 3d power of 3, and hence 3 is called the cube, or 30, root of 27. . 262. The square root of a quantity may be denoted by this character, v called the radicai sigii, placed before it, and the other roots by the same sign, with the index of the root placed over it, or by fractional indices placed on the right hand. Thus 19, or gê denotes the square root of 9, 27, or 273, denotes the cube root of 27, and v16, or 164 denotes the 4th root of 16. The latter method of denoting roots is preferable, inasmuch as by it we are able to denote roots and powers at the same time. Thus 85 signifies that 8 is raised to the second power, and the cube root of tltat power extracted, or that the cube root of 8 is extracted, and this root raised to the ed power: that is, the numerator of the index denotes the power, and the denominator the root of the number ove! which it stands. 263. Although every number must have a root, the roots of but very few numbers can be fully expressed by figures. We can however by the help of decimals approximate the roois of all sufficiently near for all prac. rical purposes. Such roots as cannot be fully expressed by figures are denominated surds, or irrational numbers. 264. The least possible root, which is a whole number, is 1. The square of 1 is ( 1 1=) 1, which has one figure less than the number em ployed as factors; the cube of 1 is (1»<l»<l=) 1. Iwo bgares less than the nuinber employed as factors, and so on. The least root consisting of i wo figures is 10, whose square is (10* 10=) 100, which has one figure less than the number of figures in the factors, and whose cube is (10 * 10 * 10–) 1000, two figures less than the number in the factors; and the game may be shown of the least roots consisting of 3, 4, &c. figures. Again, the greatest root consisting of only one fignire is 9, whose square is (9*9 =) 81, which has just the number of figures in the factors, and whosecube is (9*9*95) 729 just equal to the number of figures in the factors; and the greatest root consisting of two figures is 99, whose square is (99 99=) 9301, &c. and the same may be shown of the greatest roots consisting of 3, 4, &c. figures. Hence it appears that the number of figures in the con. tinued product of any number of factors cannot exceed the number of fig. teres in those factors; nor fall short of the number of figures in the fac. tors by the number of factors, wanting one. From this it is clear that a square number, or the second power, can have but twice as as its root, and only one less than twice as many; and that the Third power can have only three times as many figures as its root, and only iwo less than three times as many, and so on for the higher powers ; Therefore, 265. To discover the number of figures of which any root will consist: RULE.--Beginning at the right hand, distinguish the given number into portions, or periods, by dots, each portion consisting of as many figures as are denoted by the index of the root: by the number of dots will be shown the number of figures of which the root will consist. EXAMPLES. 1. How many figures in the 2. How many figures in the square, cube, and biquadrate square and cube root of 68101 root of 348753421 ? 2.1416? 3487 5342 i square root 5. 681012. 1 4 1 6 square 5 3487534 2 i cube root 3. 681012. 141 600 cube 4 348752 4 2 1 biquadrate 3. In distinguishing decimals begin at the separatrix and proceed lowards the right hand, and if the last period is incomplete, complete it by annexing the requisite number of i ciphers. EXTRACTION OF THE SQUARE ROOT. ANALYSIS. 206. To extract the square root of a given number is to find a punt. ber, which multiplied by itself, will produce the given oumber, or is is to sad the length of tbe side of a square of which the given pumber expregoes the area. • 1. 11 529 feet of boards be laid down in a square form, what will be the length of the sides of the square? or, in other words, what is the square root of 529? From what was shown [264 we know the root must consist of tivo figa ures, in as much as 529 consists of two periods. Now to understand the method of ascertaining these two figures, it may be well to consider how the square of a root consisting of two figures is formed. For this pur pose we will take the number 23 ani! 23 square it. By this operation it appears that the square of a number consisting of tens and units is made up of the 3 square of units square of the units, plus twice the pro6 iwice the product of duct of the tens, by the units plus the *6 the tens by urits. square of the tens. See this exhibited 4 square of ihe tens. in figure F. As 10>10=100, the square of the lens can never make a part of the 529 square of 23. two right hand figures of the whole square. Hence the square of the lens is always contained in the second peri529 20 ou, or in the 5 of the present example. 400 'The greatest square in 5 is 4, and its root 2; hence we conclude that the 129 tens in the root are 2 20. and 20 $20= 400. But as the square of the tens Curl never contain significant figures below hundreds we need only write the square of the figure denoting tens under the second period. From what precedes it appears that 400 of the 529 feet of boards are now disposed of in a square form, E. measuring 20 feet on each 20 st. side, and that 129 feet are to be added to this square in such manner as not to alter its form ; and in order to do this the additions must be made upon two sides of the square, E 20 20=40 feet. Now if 129, the number of feet io 20 be added, be divided by 40, the length of the 20 additions, or dropping the cipher and 9, 12 be divided by 4 the quotient will be the width of 400 ft. the additions; and as 4 in 12 is bad 3 tiines. we conclude the addition will be 3 feet wide, and 40% 3=120 feet, the quantily added upon the two sides. But since these additions are no longer than the sides of the square, A, there must be a deficiency ac the corner, as exhibited in F, whose sides 20 f. 3ft. are equal to the width of the additions, -la 20 >3566 or 3 feet, and 3>3=9 feet, required to oll out the corner, so as to co : the square. The whole operation may be arranged 13 on the next page, where i will be seen that we first find the root of the greatest square in the left hand 2020400 le period, place it in the form of a quotieri, po suhtract the 8uare from the period and to the remainder bring down the next period, which we divide, omitting the right hand figure, by double the rooi, 23 ft. and place the quotient for the second fig. ure of the root; and the square of this 20 ft. |