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If 4 M.

8,91. {100,29da.} answer £.305 0 8,2

19,5da.} 8,91. {

2 If 2 persons receive 4,625s for one day's labour, how much should 4 persons have for the work of 10,5 days? answer 41 17s 1d. 3 If 16s 4d. be the porterage of 5,25 hundred weight for 20 miles; what must be paid for carrying 17,75 hundred weight 7,5 miles ? answer 11 Os 8d. 4 How many men should reap 417,6 acres in 12 days, when 5 persons cut down of that quantity in half the time?

answer 20 men.

5 Suppose the interest of 76,941. for 9,5 months to be 15,251. what principal will gain 6l. in 12,75 months? answer 221 11s id.t 6 When 12 oxen graze down 16,25 acres, in 20 days; how much, of like pasture, would suffice 24 such cattle for 100 days? answer 162,5 acres. 7 What money, at 3 per cent. per annum, will clear 38/ 10s. in a year and a quarter?

answer 8807. 8 A cellar which is 22,5ft. long, 17,3 wide, and 10,25 deep, being dug in 21 days, by six men, working 12,3 hours a day; how many days, of 8,2 hours, should 9 men take to excavate one which measures 45, by 34,6 by 12,3 feet? answer 12 days.

INVOLUTION; OR THE RAISING OF POWERS.

A

POWER is the product arising from multiplying any given number into itself continually a certain number of times; thus,

2x2 4 is the second power, or square of 2. 2×2×2= 8 the third power, or cube of 2.

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2× 2×2×216 the fourth power of 2, &c.

The number denoting the power is called the index, or the exponent of that power.

If two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factors; thus,

2x24 the square of 2; 4x4-16=4th power of 2 ; and 16x16-256-8th power of 2, &c.

N

TABLE

[blocks in formation]

16384

655361

262144

6th power

5th power

Roots

Squares

Cubes

4th power

4/16 64 256 1024 4096 5125 125 625 3125 15625 78125 3906251 1953125 636 216 1296 7776 46656 279936] 1679616] 10077696 749 343 240116807|117649 823543| 5764801 40353607 864/512/4096/32768/262144|2097152|16777216|134217728 9811729656159049 531441 4782969|43046721/387420489

EXAMPLES.

1 What is the fifth power of 7?
~7x7x7x7x7-16807-fifth power.

2 What is the third power or cube of 35 ?

3. What is the fourth power of ?
4 What is the fifth power of,029 P
5 What is the sixth power of 5,03 ?

answer 42875

81 736

,000000020511149

answer 10190,005304479729 17857366818

6 What is the eighth power of 3 P

EVOLUTION: OR THE EXTRACTING OF ROOTS.

HE root of any number, or power, is such a number, as, being multiplied into itself a certain number of times, will produce that power. Thus 2 is the squareroot of 4, because 2×2-4; and 4 is the cube root of 64, because 4X4 X4-64, and so on.

T

THE SQUARE ROOT.

HE square of a number is the product arising from that number multiplied into itself.

Extraction of the square root is the finding of such a number. as, being multiplied by itself, will produce the number proposed.

RULE.

1. Distinguish the given number into periods of two figures each, beginning at the units place, or decimal point; and when the decimal does not consist of an even number of figures, annex a cypher; and equal to the periods of whole numbers and decimals respectively will be the places of each in the root.

2. Deduct from the first period the greatest square it contains, setting the root thereof as a quotient figure, and doubling it for a divisor; and bring down the second period to the remainder, for a dividual.

3. Try how often the said divisor, with the resulting figure of this trial thereto annexed, are contained in the dividual, and set this resulting figure to both the divisor and root; then multiply and subtract as in division, and bring down the next period.

4. Double the ascertained root for a new divisor, and repeat the process to the end.

PROOF.

Square the root, adding in the remainder (if any) which will equal the number given.

EXAMPLES.

1 What is the square root of 30138,696025?

999999

30138,696025(173,605 answer

1

27)201

189

343)1238
1029
3466)20969
20796

347205) 1756025
1736025

0

Note. When one more than half the figures of the root are found, the rest may be obtained by working as in contracted division of decimals.

2 Required the square root of 14876,2357 P

14876,2357(121,968175

1

22)48

44

241)476

241.

2429)23523
21861

24386)166257
146316

24392)19941(8175
427

183

13

1

3 Required the square-root of 5499025 ?
4 What is the square-root of 74770009 ?
5 What is the square-root of 368863 ?
6 What is the square-root of 3271,4007 P
7 What is the square-root of 2,2710957 ?
8 What is the square-root of 10?

facit 2345 answer 8647 607,34092+ 57,19+ 1,50701+ 3,162277+

9 What is the square-root of ,0003272481 P01809 10 Required the side of a square acre of land ?

facit 12,649+per. 11 A certain number of men gave 30s 1d. for a charitable purpose; each man giving as many pence as there were men: query the number ?

answer 19 men.

12 If a circular pipe of 1,5 inches diameter fill a cistern in 5 hours; in what time would it be filled by one of 3,5 inches diameter ?

answer 55min. 6sec.

13 If 484 trees be planted in a square orchard, how many must be in a row each way ?

answer 22

Note 1. The square of the longest side of a right angled triangle is equal to the sum of the squares of the other two sides; and consequently the difference of the square of the longest, and either of the other, is the square of the remaining side.

2. The square root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator, for a numerator, and of the denominator, for a denominator. Ifit be a surd, reduce it to its equi valent decimal, &c.

3. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted as before.

14 The wall of a fortress is 17 feet high, which is surrounded by a moat 20 feet in breadth; query the length of a ladder to reach from the outside of the moat to the top of the wall? answer 26,2 feet. 15 A line of 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad; the height of the wall is required?

answer 26,83+yards.

16 Suppose a ladder, 60 feet long, be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot, will reach window 23 feet high on the other side; what breadth was the street of? answer 102,64 feet.

6849

9216

6192

17 What is the square-root of 2944 ? 18 What is the square-root of 7056? 19 What is the square-root of 3168 20 What is the square-root of 3736 21 What is the square-root of 1719 ? 22 What is the square-root of 7614

THE CUBE ROOT.

answer

,71528

64

41

8,7649+

HE Cube of a number is the product of that number multiplied into its square.

Extraction of the cube root is the finding of such a number, as, being multiplied into its square, will produce the number proposed.

RULE.

First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal

N 2

point:

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