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12. Seven eighths of a certain number exceeds four

105

son's legacies £257 3s. 4d.: what was the widow's share? Ans. £635 103gd.

15. A man died, leaving his wife in expectation of an heir, and in his will ordered, that if it were a son, of the estate should be his, and the remainder the mother's; but if a daughter, the

mother

fifths, by 6; what is that should have, and the daugh

number?

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ter but it happened that she had both, a son and a daughter, in consequence of which the mother's share was $2000 less than it would have been if there had been only a daughter; what would have been the mother's portion, had there been only a son?

Ans. $1750.

REVIEW.

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6. When are fractions said to have a common denominator?

7. What is the common multiple of two or more numbers ?-the least common multiple ?-a prime number?-the aliquot parts of a number?-a perfect number? Explain.

8. What is denoted by a vulgar fraction (129)? How is an improper fraction changed to a whole or mixed number (216)?—a whole or mixed number to an improper frac tion ?

9. How is a fraction multiplied by a whole number (219) ?-divided by a whole number?

10. How would you multiply whole number by a fraction (222) 1 -a fraction by a fraction?

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 cominon 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)?-10 the least common denominator?

16. How are fractions of a higher denomination changed to a lower denomination (213)-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 (93), 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 twice into itself, it is said to F be cubed, or because it is employed 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 ront. Thus the cube root of A C E27, is A=3.

255. Again, if the cube, D, be multiplied by its root, A, the product (27×3), 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

E

A

D

B

(3×3×3×3=81), it is raised to the fourth power, or biquadrate, of which the iven number is called the fourth root.

given

256209.

INVOLUTION.

107

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 (243×3=), 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 firs1 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 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 de-. note 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

32=3X3

33=3x3x3

3, first power of 3, the root.
-9, second power, or square of 3.
27, third power, or cube of 3.

34-3×3×3×3=81, fourth power, or biquadrate of 3.

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

TABLE.

41 51 61
251

71

81 9*

361

491

641

31

Roots, or 1st
1 2 3
powers,
Squares, or 2d powers, |1| 4|_9!
Cubes, or 3d powers,
Biquadrates, or 4th p.

161 |1| 8|27| 64| 125 2161 343 512 729 1|16|81| 256| 625 12961 24011 40961 6561 Sursolids, or 5th powers, |1|32|243|1024| 3125| 7776| 16807 32768 59049 Square cubes, or 6th p. |1,64729,4096|15625|46656|117649|262144|531441

2. Evolution.

ANALYSIS.

261. The method of ascertaining, er extracting the roots of numbers; or powers, is called Evolution. The root of a number, or power, is a num 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.

3

27, or

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*, denotes the cube root of 27, and 16, 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, 8 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 belp of decimals approximate the roots of all sufficiently near for all prac tical 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

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