10. What sum of money son's legacies £257 3s. 4d.: is that whose 3d part, 4th part what was the widow's share ? and 5th part are $94 ? Ans, £635 1043d. Ans. $120. 15. A man died, leaving 11. If to my age there added be his wife in expectation of an One half, one 3d, and 3 times three, Six score and ten their sum will be; heir, and in his will ordered, What is my age ? pray show it me. that if it were a son, of the Ans. 66 years. estate should be his, and the 12. Seven eighths of a cer- remainder the mother's; but tain number exceeds four if a daughter, the mother fifths, by 6; what is that should have 3, and the daughnumber? ter }; but it happened that 13. What number is that she had both, a son and a from which if you take of , daughter, in consequence of and to the remainder add 76 which the mother's share was of z/, the sum will be 10? $2000 less than it would have Ans. 10,!,91 been if there had been only a 14. A father gave To of his į daughter; what would have estate to one of his sons, and been the mother's portion, had g of the residue to another, there been only a son? and the surplus to his relict Ans. $1750. for life; the difference in the 2240 REVIEW. 1. What are fractions? Of how 6. When are fractions said to have many kinds are fractions? In what a common denominator ? do they differ? 7. What is the common multiple 2. How is a vulgar fraction ex of two or more numbers ?-the least pressed ? What is denoted by the common multiple ?-a prime numdenominator (22)? By the numera ber?—the aliquot parts of a numtor? ber?-a perfect number? Explain. 3. What is a decimal fraction ? 8. What is denoted by a vulgar How is it expressed ? How is it fraction (129)? How is an impropread ? How may it be put into the er fraction changed to a whole or form of a vulgar fraction? mixed number (216)?-a whole or 4. What is a proper fraction ?- mixed number to an improper frac. an improper fraction? What are tion ? the terins of a fraction? What is a 9. How is a fraction multiplied by compound fraction ?-a mixed num a whole number (219) ?-divided by ber a whole number? 5. What is meant by a common 10. How would you multip!y a divisor of two numbers ?-by the whole number by a fraction (222) ? greatest common divisor i -a fraction by a fraction ? 11. How would you divide a a common denominator (239) ?-10 whole number by a fraction (225) ? the least common denominator ? - a fraction by a fraction ! 16. How are fractions of a higher 12. How may you enlarge the denomination changed to a lower terms of a fraction (229) ? How di- denomination (243)?-into integers minish them? of a lower? -a lower denomination 13. How would you find the great to a higher ?-into integers of a est common divisor of two numbers ? higher ? How reduce a fraction to its lowest 17. Is any preparation necessary terms ? in order to add 'fractions (249) ?14. How would you find a com- why must they have the same demon multiple of two numbers (236) ? nominator ? How are they added ? the leasi common multiple ? How is subtraction of fractions per 15. How are fractions brought to formed? How the rule of three ? SECTION VIII. POWERS AND ROOTS. 1. Envolution. ANALYSIS. 253. Let A represent a line 3 feet long; if this length A bè multiplied by itself, the product (3X3), 9 feet, is the area of the square, B, which measures 3 feet on every side. B Hence, if a líne, or a number, be multiplied by itself, it is said to be squared, or because it is used twice as a factor, it is said to be raised to the second power; and the lino which makes the sides of the square is called the first power; the root of the square, or its squure root. Thus, the square root of B=9, ís A33. 254. Again, if the square, B, bc multiplied by A B its root, A, the product (9X3=), 27 feet, is the volume, or content, of the cube, A CE, which measures 3 feet on every side. Hence, if a line or a number be multiplied iwice into itself, it is said to F be cubed, or because it is employed 3 times as a factor (3X3X3=27), it is said to be raised to the third power, and the line or number which show's the dimensions of the cube, is called its cube root, Thus the cube root of ACE=27, is A=3. E 255. Again, if the cube, D, be multiplied by its root, A, the product (27*3=), 81 feet, is thie content of a parallelopipedon, Ä C E, whose D 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 multiplied 3 times into itself, or employed four times as a factor (3X3X3X331), it is raised to the fourth power, or biquadrate, of which the given number is called the fourth root. 256. Again, if the biquadrate, D, be multiplied by its root, A, the product, (81X3=) 243, is the content of a plank, equal to 9 cubes, A CE, laid down in a square form, and called the 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 produci (243X3), 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 that ihe forms of the 3d, 4th and 5th powers are continually repeated; that is, the 3d, 6th, 9th, &c. powers will be cubes, the 4th, 7th, 10th, &c. parallelopipedons, and the 5th, 8th, 11th, &c. planks. The raising of power of numbers is called INVOLUTION. 259. The number which denotes the power to which another 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 denoted by the index of the power, thus :: 3-3 33, first power of 3, the root. 32–3X3 =9, second power, or square of 3. 33=3X3X3 =27, third power, or cube of 3. 3433X3X3X3=81, fourth power, or biquadrate of 3. QUESTIONS FOR PRACTICE. 1. What is the fifth power 2. What is the second of 6? power of 45 ? 6 Ans. 2025. 6 3. What is the square of 36 2d power. 0.25 feet (121)? Ans. 0.0625 ft. 6 4. What is the square of 216 3d power. Ans. fin. 6 5. What is the cube of 11, or 1.5? 1296 4th power. Ans. 27-3, or 3.375. 6 6. How much is 44 ? 62 ? Ans. 7776 5th power. 83 ? 75 ? 114 ? 2010 ? inch? 260. The powers of the nine digits, from the first to the sixth inclusive, are exhibited in the following TABLE. (Roots, or 1st powers, 111 21 31 41 51 61 7 기 81 99 Squares, or 2d powers, 111 41 9! 16|| 251 361 491 64| 81 Cubes, or 3d powers, 111 81 271 641 125 2161 3431 5121 729 Biquadrates, or 4th p. 11|16|81| 256 625) 12961 2401 4096 6561 Sursolids, or 5th powers, 11|32|243|1024| 3125] 7776| 16807| 32768| 59049 Square cubes, or 6th p. 11164 729|4096|15625146656|117649|262144|531441 2. Evolution. a num ANALYSIS. 261. The method of ascertaining, or extracting the roots of numbers, or powers, is called Evolution. The root of a number, or power, is ber, which, multiplied by itself continually, a certain number of times, will produce that power, and is named from the denomination of the power, 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 3d, root of 27. 262. The square root of a quantity may be denoted by this character „, called the radical sign, 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, V9, or 9$, denotes the square root of 9, 327, or denotes the cube root of 27, and $16, or 167, 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, signifies that 8 is raised to the second power, and the cube root of that power extracted, or that the cube root of 8 is extracted, and this root raised to the second power ; that is, the numerator of the index denotes the power, and the denominator the root of the number over which it stands. 275 88 263. 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 roots of all sufficiently near for all practical 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 (1x1=) 1, which has one figure less than the number employed as factors; the cube of 1 is (1X1X1=) 1, two figures less than the number empioyed as factors, and so on. The least root consisting of iwo figures is 10, whose square is (10x10=) 100, which has one figure less than the number of figures in the factors, and whose cube is (10x lox 10_) 1000, two figures less than the number in the facwrs; and the same may be shown of the least roots consisting of 3, 4, &c. figures. Again, the greatest root consisting of only one figure, is 9, whose square is (9X9=) 81, which has just the number of figures in the factors, and whose cube is (9X9 X9=) 729, just equal to the number of figures in the factors; and the greatest root consisting of two figures, is 99, whose square is (99X993) 5801, &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 continued product of any number of fuctors cannot exceed the number of figures in those factots ; nor full short of the number of figures in the factors 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 many figures as its oot, and only one less than twice as many; and that the third power can ave only three times as many figures as its root, and only two less than hree 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 ? 34875342 square root 5. 681012. 1416 square 5. 3487534 2 i cube root 3. 681012141600 cube 4. 34875242 i biquadrate 3. In distinguishing decimals, begin at the separatrix and proceed iowards the right hand, and if the last period is incomplete, complete it by annexing the requisite number of ciphers. EXTRACTION OF THE SQUARE ROOT. ANALYSI8 266. To extract the square root of a given number is to find a pumber, which, multiplied by itself, will produce the given number, or it is to find the length of the side of a square of which the given number expresses the area. 10 |