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A General Rule for extracting the Roots of all Powers,
1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs.
2. Find the first figure of the root by trial, and subtract its power from the left hand period of the given number.
3. To the remainder bring down the first figure in the next period, and call it the dividend.
4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor.
5. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root.
6. Involve the whole root to the given power, and subtract it (always) from as many periods of the given number as you have found figures in the root.
7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor as before, and in like marner proceed till the whole be finished.
NOTE.—When the number to be subtracted is greater than those periods from which it is to be taken, the last cuotient figure must be taken less, &c.
1. Required the cube root of 135796,744 by the above general method.
135796744(51,4 the root.
132651=2d subtrahend. 7803) 31457=2d dividend.
5 X5 X3=75 first divisor.
514 X 514 X 514=135796744 3d subtrahend. 2. Required the sursolid or 5th root of 6436343.
23 x 23 x 23 x 23 x 23=6436343 subtrahend. NOTE.--The roots of most powers may be found by the square and cube roots only; therefore, when any ereu power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which reduces it to half the given power, then the square root of that power
reduces it to half the same power; and so on, till you come to a square or a cube.
For example: suppose a 12th power be given; the squaro root of that reduces it to a 6th power: and the square roof of a 6th power to a cube.
3. What is the biquadrate, or 4th root of 199871733761
Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464.
Ans. 48. 5. Extract the square, biquadrate, or Sth root of 72139 95789338836.
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ALLIGATION, IS the method of mixing several simples of different qualities, so that the composition may be of a mean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate.
ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials.
RULE. As the whole composition : is to the whole value : : so berekend is any part of the composition : to its mean price.
bu. cts. $cts. bu. $ cts. bu.
54)25,38(,47 Ans. 54 25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth ?
Ans. $1,20 cts. 3. A tobacconist mixed 36 lb. of tobacco, at Is. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per 1b.; what is the price of a pound of this mixture ?
Ans. ls. 8d. 4. A grocer mixed 2 C. of sugar at 56s. per C, and 1 C. at 43s. per C. and 2 Č. at 50s. per C. together; I demand-the price of 3 cwt. of this mixture ? Ans. £7 13s.
5. A wine merchant mixes 15 gallons of wine at 4s. ed.
per gallon, with 24 gallons at 6s. 8d. and 20 gallons at 6s. 3d. ; what is a gallon of this composition worth?
Ans. 5s. 10d. 21 grs.
6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort # at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. and he would mix an equal quantity of each together; 1 demand the price of 3) cwt. of this mixture ?
Ans. $34 12 cts. 5 m.
1 7. A goldsmith melted together 5 lb. of silver bullion, A of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine; wort pray what is the quality or fineness of this composition? per
Ans. 6 oz. 13 pwt. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together ; what is the quality or fineness of this mass?
20 Ans. 19 carats fine.
lity ALLIGATION ALTERNATE,
12 IS the method of finding what quantity of each of the As ingredients whose rates are given, will compose a mixture 3. of a given rate; so that it is the reverse of Alligation Me
per dial, and may be proved by it. CASE I.
4. When the mean rate of the whole mixture, and the rates of all the ingredients are given, without any limited quan. tity.
A RULE. 1. Place the several rates, or prices of the simples, be 48. a ing reduced to one denomination, in a column under each ther, other, and the mean price in the like name, at the left hand
muc 2. Connect, or link the price of each simple or ingredi- An ent, which is less than that of the mean rate, with one of any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num. 6. ber of the less.
cts. 3. Place the difference, between the mean price (or mix, at 8 ture rate) and that of each of the simples, opposite to the mix rates with which they are connected.
4. Then, if only one difference stands against any rate, It will be the quantity belonging to that rate, but if there be several, their sum will be the quantity,
1. A merchant has spices, some at 9d. per lb. some at Is. some at 2s. and some at 2s. 6d. per lb. how much of each sort must he mix, that he may sell the mixture at ls. 8d. per pound? d. 16. d.
d. 16. 9 10 at 97
4 d. 12 4 12 1 Gives the d. | 12 10 20) 24 8 24 | Answer; or
20 24 ) 11 30 11 30
30 2. A grocer would mix the following qualities of sugar; riz. at 10 cents, 13 cents, and 16 cents per lb. ; what quantity of each sort must be taken to make a mixture worth 12 cents per pound ? Ans. 5 lb. at 10 cts. 2 lb. at 13 cts. and 2 lb. at 16 cts. per lh.
3. A grocer has two sorts of tea, viz. at 9s. and at 15s. per lb. how must he mix them so as to afford the composiion for 12s. per lb. ?
Ans. He must mix an equal quantity of each sort.
Ans. 2 of each of the first three sorts, and 9 of the last.
5. It is required to mix several sorts of rum, viz. at 58. Side 7s. and 9s. per gallon, with water at per gallon, togeed ther, so that the mixture may be worth 6s. per gallon; how
much of each sort must the mixture consist of ?
1 gal. water.
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