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2. A

grocer would mix teas at 125. Ios. and 6s. with 20lb. at 45. per pound; how much of each sort must he take to make the composition worth Ss. per lb ?

Ans. 20lb. at 45. 10 at 6s. 10 at 1os. and 20 at 125. 3. How much gold of 15, of 17 and of 22 carats fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine ?

Ans. 5oz. of 15 carats fine, 5 of 17, and 25 of 22.

INVOLUTION.

A POWER is a number produced by multiplying any given number continually by itself a certain number of times.

Any number is itself called the first power ; if it be multiplied by itself, the product is called the second power, or the square ; if this be multiplied by the first power again, the product is called the third power, or the cube ; and if this be multiplied by the first power again, the product is called the fourth power, or biquadrate ; and so on ; that is, the power is denominated from the number, which exceeds the multiplications by 1.

Thus, 3 is the first power of 3.

3X3= 9 is the second power of 3. 3X3X3=27 is the third power of 3. 3X3X3X3=81 is the fourth power of 3. &c.

&c. And in this manner is calculated the following table of powers.

TABLE

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4th Pow. I

16

81

256

625

1266

2401

4096

6561

sth Pow. I

321

243

1024

31251

7776

16807

32768

59049

6th Pow. 1

641

729

4096

15625

46656

1176491

262144

531441

TABLE of the first twelve Powers of the 9 DIGITS,

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19th Pow. I 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1 oth Pow. 1 1024 59049 1048576 9765625 604661761 282475249 1073741824 3486784401

I ith Pow. 1 2048 177147 4194304 48828125 362797056 1977326743 8589934592 31381059609

Ist Pow.

2d Pow.

3d Pow.

NOTE II

12th Pow. 1096 531441 16777216 244140625 2176782336,1384128720163719476736 282420536481 NOTE 1. The number, which exceeds the multiplications by 1, is called the index, or exponent, of the power; so the index of the first power is 1, that of the second power is 2, and that of the third is 3, &c. NOTE 2.

Powers are commonly denoted by writing their indices above the first power : so the second power of

3 inay be denoted thus 3*, the third power thus 3', the fourth power thus 3*, &c. and the sixth power of 503 thus 5030

Involution is the finding of powers ; to do which we have evidently the following

RULE.

Multiply the given number, or first power, continually by itself, till the number of multiplications be i less 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 raising each of their terms to the power required. And if a mixed number be

proposesi,

* Note. The raising of powers will be sometimes shortened by working according to this observation, viz. whatever two or more powers are multiplied together, their product is the power, , whose index is the sum of the indices of the factors ; or if a power be multiplied by itself, the product will be the power, whose index is double of that, which is multiplied : so if I would find the sixth power, I might multiply the given number twice by itself for the third power, then the third power into itself would give the sixth power ; or if I would find the seventh power, I might first find the third and fourth, and their product would be the seventh ; or lastly, if I would find the eighth power, I might first find the second, then the second into itself would be the fourth, and this into itself would be the eighth.

proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule.

EXAMPLES,

Ans. 2025, 1. What is the second power of 45 ?

Ans. '000729. 2. What is the square of '027?

Ans. 42.875 3. What is the third power of 3'5 ? 4. What is the fifth power of .029 ?

Ans. '000000020511149. 5. What is the sixth power of 5'03 ?

Ans. 16196*005304479729, 6. What is the second power of j?

Ans.

EVOLUTION.

The Root of any given number, or power, is such number as, being multiplied by itself a certain number of times, will produce the power ; and it is denominated the first, second, third, fourth, &c. root, respectively, as the number of multiplications made of it to produce the given power

is 0, 1, 2, 3, &c. that is, the name of the root is taken from the number, which exceeds the multiplications by. I, like the name of the

power

in involution,

NOTE I. The index of the root, like that of the power in involution, is 1 more than the number of multiplications, necessary to produce the power or given number. NOTE 2.

Roots are sometimes denoted by writing before the power, with the index of the root against it :

3

so the third root of 50 is ✓ 50, and the second root of it is a 50, the index 2. being omitted, which index is always understood, when a root is named or written without one, But if the power be expressed by several numbers with the sign + or -, &c. between them, then a line is drawn

from

1

from the top of the sign of the root, or radical sign, over
all the parts of it : so the third root of 47—15 is
3
V 47 — 15.

And sometimes roots are designed like
powers, with the reciprocal of the index of the root above
the given number. So the 2d root of gis 37; the 2d root of 50
is sož ; and the shird root of it is 503; also the third root of
47—15 is 47—15| And this method of notation has
justly prevailed in the modern algebra ; because such roots,
being considered as fractional powers, need no other di-
'rections for any operations to be made with thein, than
those for integral powers.
Note

3.

A number is called a complete power of any kind, when its root of the same kind can be accurately extracted ; but if not, the number is called an imperfect power, and its root a surd or irrational number : so 4 is a complete power of the second kind, its root being 2 ; but an imperfect power of the third kind, its root being a surd number.

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Evolution is the finding of the roots of numbers either accurately, or in decimals, to any proposed extent.

The power is first to be prepared for extraction, or evolution, by dividing it from the place of units, to the left in integers, and to the right in decimal fractions, into periods containing each as many places of figures, as are denominated by the index of the root, if the power contain a complete number of such periods : if it do not, the defect will be either on the right, or left, or toth; if the defect be on the right, it may be supplied by annexing cyphers, and after this, whole periods of cyphers may be annexed to continue the extraction, if necessad ry; but if there be a defect on the left, such defective period must remain unaltered, and is accounted the first period of the given number, just the same as if it were complete:

Now

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