2. A grocer would mix teas at 12s. Ios. and 6s. with 20lb. at 4s. per pound; how much of each sort must he take to make the composition worth 8s. per lb ? Ans. 20lb. at 45. 10 at 6s. 10 at 10s. and 20 at 128. 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. 3×3×3 27 is the third power of 3. And in this manner is calculated the following table of TABLE of the first twelve POWERS of the 9 DIGITS, powers. TABLE 2d Pow. Ist Pow. I I 4 3 2 4 3 9 16 25 36 49 64 81 6th Pow. I 64 729 4096 15625 46656 117649 262144 531441 I 128 2187 16384 78125 279936 823543 2097152 4782969 3d Pow. 4th Pow. 5th Pow. 7th Pow. 8th Pow. 10th Pow. 1 1024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401 NOTE 1. The number, which exceeds the multiplications by I, 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 may be denoted thus 3*, the third power thus 33, the fourth power thus 3*, &c. and the sixth power of 503 thus 503. 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 proposed, * 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 THE ROOT of any given number, or power, is such a 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 1, like the name of the power in involution, NOTE 1. The index of the root, like that of the power in involution, is I 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✓ 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 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 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 3 is 3; the 2d root of 50 is 50 I I ; and the third root of it is 503; also the third root of I 3 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 then, 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. 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 both; 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 necessary; 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 |