10. What sum of money | son's legacies £257 38. 4d.: is that whose 3d part, 4th part what was the widow's share?

REVIEW.

11. How would you divide a whole number by a fraction (225)? -a fraction by a fraction?

12. How may you enlarge the terms of a fraction (229)? How diminish them?

13. How would you find the greatest common divisor of two numbers? How reduce a fraction to its lowest terms?

14. How would you find a common multiple of two numbers (236)? the least common multiple? 15. How are fractions brought to

a common denominator (239)?-to the least common denominator?

16. How are fractions of a higher denomination changed to a lower denomination (243)?-into integers of a lower?-a lower denomination to a higher ?-into integers of a higher?

17. Is any preparation necessary in order to add fractions (249)?— why must they have the same denominator? How are they added? How is subtraction of fractions performed? 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 be multiplied by itself, the product (3×3), 9 feet, is the area of the square, B, which measures 3 feet on every side. Hence, if a line, 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 line which makes the sides of the square is called the first power; the root of the square, or its square root. Thus, the square root of B-9, is A-3.

254. Again, if the square, B, be multiplied by its root, A, the product (9x3), 27 feet, is the volume, or content, of the cube, A C E, which measures 3 feet on every side. Hence, if a line or a number be multiplied twice into itself, it is said to F be cubed, or because it is employed 3 times as a factor (3x3x327), it is said to be raised to the third power, and the line or number which shows the dimensions of the cube, is called its cube root. Thus the cube root of A C E 27, is A=3.

255. Again, if the cube, D, be multiplied by its root, A, the product (27x3=), 81 feet, is the content of a parallelopipedon, A CE, whose length is 9 feet, and other dimensions, 3 feet each way, equal to 3 cubes, A C E, placed end to end. Hence, if a given number be multiplied 3 times into itself, or employed four times as a factor

(3x3x3x3=81), 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 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 firs 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 the forms of the 3d, 4th and 5th powers are continually repeated; that is, the 3d, 6th, 9th, &c. powers will 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:

33

32=3X3

=3, first power of 3, the root.

9, second power, or square of 3. =27, third power, or cube of 3.

34-3X3X3X3-81, fourth power, or biquadrate of 3.

260. The powers of the nine digits, from the first to the sixth inclusive, are exhibited in the following

Biquadrates, or 4th p.

|1|16|81| 256 625 1296 2401 4096 6561 Sursolids, or 5th powers, |1|32|243|1024| 3125| 7776| 16807| 32768| 59049 Square cubes, or 6th p. |1|64|729|4096|15625|46656|117649|262144|531441|

2. Evolution.

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 a number, 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, 9, or 94, denotes the square root of 9, 27,

4

3.

or

27, denotes the cube root of 27, and 16, or 16, 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, 83 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.

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 (IX) 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 employed as factors, and so on. The least root consisting of two 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 (10x10x 10) 1000, two figures less than the number in the factors; 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 (9×9 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 (99×99) 9801, &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 factors cannot exceed the number of figures in those factors; nor fall 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 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 two 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 square, cube, and biquadrate root of 348753421 ?

348753421 square root 5. 348753421 cube root 3. 848752421

biquadrate 3.

2. How many figures in the square and cube root of 68101 2.1416?

681012. 1416 square 5. 681012.141600 cube 4.

In distinguishing decimals, begin at the separatrix and proceed towards the right hand, and if the last period is incomplete, complete it by annexing the requisite number of ciphers.

EXTRACTION OF THE SQUARE ROOT.

ANALYSIS.

266. To extract the square root of a given number is to find a number, 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