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$4353564.67 Answer. 2. A country gentleman wishing to buy some oxen, meets with a person who had 20; he demanded the price of them, and was answered $32 a-piece; the gentleman offers him $30 a-piece, and he would buy all : the other tells him it could not be taken, but if he would give what the last ox would come to, at 1 cent for the first, and doubling to the last, he should have all, which he agreed to.- I wish to know how much he paid for them.

Ans. 85242.88. 3. A man agreed with his neighbour for a team of 5 horses; and was to give 9 cents for the first horse, 9 times as much for the second, increasing the price of each horse in a nineold ratio—What was the price of the team ?

Ans. $664.29. 4. If a man was to work 23 days for the following wages, viz. at 1 mill for the first day's work, 3 for the second, 9 for the third, and so on, increasing each day's wages in a threefold proportion-Required the amount of his wages.

Ans. $47071589.413. 5. The first term of a decreasing geometrical series is 2048, the ratio ž, and the number of terms 12—Required the sum of the series.

Ans. 4095,

INVOLUTION,

OR THE METHOD OF RAISING POWERS.

A power is the product arising from multiplying any given numberinto itself continually a certain number of times; thus,

4x4 = 42 the 2d power or square of 4.
4x4x4 - 43 the 3d power or cube of 4.

4x4x4x4=4* the fourth power of 4. &c. The number denoting the power is called the index of that power.

RULE.

Multiply the number, continually by itself, till the number of multiplication be one less than the index of the power to be found, the last product will be the power required.

Examples. 1. What is the sixth

power

of 8? 8

64=2d power,
8

512=3d power,

8

4096=4th power,

8

32768=5th power,

8

262144=6th power. 2. Required the 4th power of 14.

Ans. 38416. 8. What is the square of 106 ?

Ans. 11236. 4. What is the cube of 97?

Ans. 912673. 5. Required the 4th power of 1.5. Ans. 5.0625.

6. A square chamber is 31 feet each way-How many square feet does it contain?

Ans. 961. 7. In a plantation 160 perches square, how many square perches ?

Ans. 25600. From the foregoing examples, we deem the doctrine of raising powers clear.

EVOLUTION,

OR THE EXTRACTION OF ROOTS.

The root of any number or power, is such a number, as, being multiplied into itself a certain number of times, will produce that number or power. Thus, 4 is the square root of 16, because 4x4 = 16; and 3 is the cube root of 27, because 3x3x3=27, and so on.

EXTRACTION OF THE SQUARE ROOT. To extract the square root is to find out such a number, as, being multiplied into itself, the product will be equal to that number.

RULE. First-Prepare the number for extraction, by pointing it off from the units place, or decimal point, into periods of two figures each; and when the decimal does not consist of an even number of figures, annex a cipher.

Secondly-Seek the greatest square number that is contained in the first point towards the left hand; place the square number under the first point, and the root thereof as a quotient figure; subtract the square number from the first point, and to the remainder bring down the next point for a dividend.

Thirdly_Double the root already found, and place it for a divisor, on the left hand of the dividend, and find how often it is contained in the dividend, exclusive of the place of units; annex the result to the quotient, and also to the divisor; then multiply by the figure last put in the quotient, subtract as in division, and bring down the next period for a new dividend.

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

Note The root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator for a new numerator, and of the denominator for a new

N

denominator; a mixt number may be reduced to an improper fraction, and the root thereof extracted as before. If the fraction be a surd, that is, a number where a root can never be exactly found, reduce it to a decimal, and extract the root from it.

The following rule, the same in substance with the foregoing, may perhaps be more easily recollected: First, to prepare the square, this do,

,
Point off the figures two by two;
Beneath the last the square next less
Put, and its root in the quotient place;
From the last period take the square,
Then the next lower period there
To the remainder must be brought;
Be this a dividend: the quote
Doubled must the divisor be
To all but units place; then see
How oft the greater holds the less,
That figure must the quote express,
And the divisor units too,
Then as in plain division do,
Thus every period one by one
We manage, and the work is done.

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

Examples. 1. What is the square root of 185511.1041?

185511.1041(430.71 Answer.
16

83,255

249

860761110

60249

86141)86141

86141

2. Required the square root of 10.

10.00000000 (3.16223 Answer, nearly.
9

61)100

61

626 3900

3756

6322)14400

12644

63242)175600

126484

Ans. 15

632448)4911600

5059584 3. What is the square root of 299182436 ? Ans. 17294. 4. What is the square root of 49491225 ? Ans. 7035. 5. Required the square root of 41370624. Ans. 6432. 6. What is the square root of 11831 ?

Ans. 109. 7. Required the square root of 10759.56022089.

Ans. 103.7283. 8. Required the square root of 2.5? Ans. 1.58114. 9. How much is the square root of 257 ? Ans. 16.0312. 10. Required the square root of 99.99. Ans. 9.9995. 11. What is the square root of 39815? 12. What is the square root of .00172? Ans. .0131148.

Further use of the Square Root. 13. If 1024 trees be planted in a square orchard, how many must be planted in a row each way? Ans. 32. 14. Required the length of a side of a square acre of land.

Ans. 69 yards, 1 ft. 8. in. + Note.-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 squares of the longest and either of the other sides, is the square of the remaining side.

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