2. If 9 bhds. of tobacco, contain 85cwt. Ogr. 3b, tare 30 per bhd. tret and cloff as ufual; What will the neat weight come to, at 6d. per ft, after deducting, for duties and other charges, £51 11s. 8d.?

Anf. £187 18s. 5d.

INVOLUTION,

Or, то

RAISE POWERS.

A Power is the product arifing from multiplying any given number into itself continually a certain number of times, thus:

3X39 is the 2d power, or square of

(3.32

(3.33

3X3X3=27 is the 3d power, or cube of

3X3X3X3=81 is the 4th power, or the biquad

(rate of 3, &c. =

34

The number denoting the power is called the Index, or the Exponent of that power; thus, the fecond power

of

5 is

25, or 52, &c.

2 X 24, the fquare of 2: 4 × 4=16=4th power of 2: 16×16=256=8th power of 2, &c.

RULE.-Multiply the given number, root, or first power continually by itfelf, till the number of multiplications be lefs than the index of the power to be found, and the last product will be the power required.

Note. Whence, because fractions are multiplied by taking the products of their numerators, and of their denominators, they will be involved by raifing each of their terms to the power required; and if a mixed number be propofed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule.

A

Second Surfolids,

128 2187 16384 78125

279936

823543

2097152

4782969

The Root is a number whofe continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th root, &c. ac cordingly, as it is, when raised to the 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16; because 4X416, and 3 is the cube root of 27, because 3X 3X3=27; and so on.

Although there is no number of which we cannot find any power exactly, yet there are many numbers, of which precife roots can never be determined. But by the help of decimals, we can approximate towards the root, to any affigned degree of exactnefs.

The roots, which approximate; are called furd roots, and those which are perfectly accurate, are called rational roots.

Roots are fometimes denoted by writing the character before the power, with the index of the root over it; thus the 3d root of 36 is expreffed 3 36, and the 2d root of 36 is

36, the index 2 being omitted when the fquare root is defigned.

If the power be expreffed by feveral numbers, with the fignor between them, a line is drawn from the top of the fign over all the parts of it; thus,

-----

3

the 3d root of 47 + 22 is √47 +22, and the 2d root 17 is 59-17, &c.

of 59

Sometimes roots are defigned like powers, with

fractional indices; thus, the fquare root of 15 is 157,

I

the cube root of 21 is 213, and the 4th root of 3720 is 37-204, &c.

The EXTRACTION of the SQUARE ROOT.

RULE 1.-Diftinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points fhew the number of figures the root will confist of.

2. Find the greateft fquare number in the firft, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in divifion) for the firft figure of the root, and the square number, under the period, and fubtract it therefrom, and. to the remainder bring down the next period for a dividend.

3. Place the double of the root, already found, on the left hand of the dividend for a divifor.

4. Seek

4

4. Seek how often the divifor is contained in the dividend, (except the right hand figure) and place the answer in the root for the fecond figure of it, and likewife on the right hand of the divifor: Multiply the divifor, with the figure laft annexed, by the figure laft placed in the root, and subtract the product from the dividend: To the remainder join the next period for a new dividend. 5. Double the figure already found in the root, for a new divifor, (or, bring down your last divifor for a new one, doubling the right hand figure of it) and from thefe, find the next figure in the root as laft directed; and continue the operation, in the fame manner, till you have brought down all the periods.

Note I If, when the given power is pointed off as the power requires, the left hand period should be deficient, it must neverthelefs ftand as the first period.

Note If there be decimals in the given number, it must be pointed both ways from the place of units: If, when there are integers, the first period in the decimals be deficient, it may be com pleted by annexing fo many cyphers as the power requires: And the root must be made to confift of fo many whole numbers and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhaufted, the operation may be continued at pleasure by annexing cyphers.

EXAMPLES.

1. Required the fquare root of 30138696025?

30138696025(173665 the root.

I

ift Divifor 27)201

189

2d Divifor 343)1238

1029

3d Divifor 3466)20969

20796

4th Divifor 347205)1736025

1736025

2. Required