2.

If y bhds. of tobacco, contain 85cwt. Oqr. 31,

9 tare 3010, per hhd. tret and cloff as usual ; What will the neat weight come to, at 6 d. per tb, after deducting, for duties and other charges, £ 51 115. 8d.?

Any: £187 18s. 5d.

INVOLUTION,

Or, TO

R A I S E POWER S.

2

A Power is the product arising from multiplying any given number into itself continually a certain number of times, tnus : 3x3=9 is the 2d power, or square of

(3. = 32

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

(3. = 33 3X3X3X3=81 is the 4th power, or the biquad

(rate of 3, &c. = 34

3 The number denoting the power is called the Index, or the Exponent of that power ; thus, the second power of 5 is 25, or 5?, &c.

2 X 2=4, the square of 2 : 4 X4=16=4th power of 2 : 16x16=256=8th power of 2, &c.

Ruls.-Multiply the given number, root, or first power continually by itself, till the number of multiplications be : less than the index of the power to be found, and the lait 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 raising each of their terms to the power required ; and if a mixed number be proposed, either reduce it to an improper traction, or reduce the vulgar fraction to a decimal, and proceed by the rule.

a

64

7/11

128

8

Square Cubes,

61

729 4096 15625 46656 117649 262144

5314411 Second Sursolids,

2187 16384

781251
279936
823543 2097152

4782969
Biquadrates, Squared ;

256 6361

65535 3906251 1679816 5764801 16777216 43046721
Cubes, Cubed,

91 512 19683 262144 1953125 10077696 40353607 134217728 387420489
Gursolids, Squared ; 10 1024 59049 1048576 9755625 60466176 282475249 1073741824 4386784454
Third Sursolids, 11102048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609
Isquare Cubes, Squared ; 12/1140961531441167772161 244140625 2176782336613841287201 68719476736 282429536481

EXAMPLES.

The Root is a number whose 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 4 X4=16, and 3

is the cube root of 27, because 3 X3 X 3=27 ; and so on.

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

The

3

The roots, which approximate ; are called surd roots, and those which are perfectly accurate, are called ran tional roots.

Roots are sometimes denoted by writing the character ✓ before the power, with the index of the root over it ; thus the 3d root of 36 is expressed V 36, and the ad root of 36 is ✓ 36, the index 2 being omitted when the square root is defigned.

If the power be expressed by several numbers, with the sign + or -- between them, a line is drawn from the top of the sign over all the parts of it; thus, the 3d root of 47 + 22 is 47 +22, and the 2d root

17 is 59—17, &c.
Sometimes roots are designed like powers,

with fractional indices ; thus, the square root of 15 is 157, the cube root of 21 is 21}, and the 4th root of

37 20 is 37 — 204, &c.

c

a

3

of 59

The Extraction of the SQUARE ROOT.

2.

Rule. 1.-Distinguish 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 shew the number of figures the root will consist of.

Fifu the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract 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 divisor.

4. Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor: Multiply the divisor, with the figure last annexed, by the figure last 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 divisor, (or, bring down your last divisor for : new one, doubling the right hand figure of it) and from these, find the next figure in the root as last directed ; and continue the operation, in the same 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 neverthcless stand as the first period.

Nire 2. 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 completed by annexing so many cyphers as the power requires : And the root must be made to consist of so many whole numbers and decimals as there are periods belonging to each ; and when the periods belonging to the given number are exhausted, the operatio: may be continucd at pleasure by annexing cyphers.

E x A M P L E S. 1. Required the square root of 30138696025?