INVOLUTION. 512. INVOLUTION is the process of finding the powers of quantities. A power of a number or quantity is the result obtained by taking that quantity a certain number of times as a factor. 513. The number from which a power is derived is called the root of that power. The first power is the root, or the number involved. The second power is the product of the root multiplied by itself once, or used twice as a factor. The third power is the root used three times as a factor; &c. 514. The index or exponent of a power is a small figure written at the right, above the root, indicating the number of times it is employed as a factor. Thus, the second power of 4 is written 42, the third power of 9 is written 9), and the fourth power of is written (). NOTE. In denoting the power of a fraction, the fraction is included in a parenthesis, in order that the exponent may be regarded as applying to the whole expression, and not to the numerator alone. When no index is written, the number itself is to be considered the first power. The second power is sometimes called the square of a number, the third power the cube, and the fourth power the biquadrate. 515. To raise a number to any required power. 2, the first power of 2, is written 2 or 2. 2 X 2 4, the second power of 2, is written 22. 2 X 2 X 2 = 8, the third power of 2, 23. 2 X2 X2 X2 16, the fourth power of 2, 24. 2 X2 X2X2 X2 = 32, the fifth power of 2, By examining the several powers of 2 in the examples, it is seen that each has been produced by taking the 2 as a factor as many times as there are units in the exponent of each power raised. Hence the RULE. — Multiply the given number into itself, till it has been used as a factor as many times as there are units in the exponent of the power to which the number is to be raised. 2 = 66 66 25. NOTE 1. The number of multiplications will always be one less than the number of units in the exponent of the power to be raised, since in the first multiplication the root is used twice; once by being taken as the multiplicand, and once more as the multiplier. NOTE 2. - A fraction is involved by involving both its numerator and its denominator. EXAMPLES Ans. 213. 1. What is the 3d power of 8 ? Ans. 512. 2. What is the 5th power of 4 ? Ans. 1024. 3. What is the 3d power of ? Ans. 37 4. What is the 4th power of 23 ? Ans. 504 5. What is the 5th power of }? 6. What is the 6th power of 5 ? Ans. 15625. 7. What is the 6th power Ans. 161314 8. What is the value of 710 ? Ans. 282475249. 9. What is the value of .0454 ? Ans. .000004100625. 516. To raise a number to any required higher power, without producing all the intermediate powers. Ex. 1. What is the 7th power Ans. 78125. OPERATION. + 2 1 2 3 2 7 + ; 5 We raise the 5 to the 2d and to the 3d power, and write above each power its exponent. Then, by adding the exponent 2 to itself, and increasing the sum by the exponent 3, we obtain 7, a number equal to the exponent of the required power; and by multiplying 25, the power belonging to the exponent 2, into itself, and the product thence arising by 125, the power belonging to the exponent 3, we obtain 78125, the required 7th power. Therefore, The product of two or more powers of the same number is that power which is denoted by the sum of their exponents. Hence, the RULE. — Multiply together two or more powers of the given number, the sum of whose exponents is equal to the exponent of the power required, and the product will be that power. NOTE. When the number to be involved contains a decimal, it is generally sufficient to retain in the result not more than six places of decimals; and the work may be accordingly contracted as in the multiplication of decimals (Art. 273). EXAMPLES 2. What is the 7th power of 8 ? 3. What is the 9th power of 7 ? 4. What is the 10th power of 6? Ans. 2097152. Ans. 40353607. Ans. 64 72*g. 5. What is the 5th power of 195 ? Ans. 281950621875. 6. What is the 6th power of } ? 7. Required the 2d power of 4698. 8. Required the 2d power of 6031. Ans. 36372961. 9. What is the 13th power of 7? Ans. 96889010407. 10. What is the 12th power of 6? 11. What is the 15th power of 9? Ans. 205891132094649. 12. What is the 4th power of 4.367 ? Ans. 363.691179+. 13. Involve the following numbers to the powers denoted by their respective exponents: (2x), 1.04', and (34)". Ans. 15722; 1.800943+; 1161331 EVOLUTION 517. EVOLUTION, or the extraction of roots, is the process of finding the roots of quantities. It is the reverse of involution. 518. The root of a quantity or number is such a factor as, being multiplied into itself a certain number of times, will produce that quantity or number. The root takes the name of the power of which it is the correlative term. Thus, if the number is a second power, the root is called the second or square root; if it is the third power, the root is called the third or cube root; if it is the fourth power, its root is called the fourth or biquadrate root; and so on. Rational roots are such as can be exactly obtained. Surd roots are such as cannot be exactly obtained. 519. Roots are usually denoted by writing the radical sign, , before the power, with the index of the root over it; in case, however, of the second or square root, the index 2 is omitted. Thus, the third root of 27 is denoted by 27, the second root of 16 is denoted by <16, and the fourth root of 3 is denoted by 1. Roots are sometimes denoted by a fractional index or exponent, of which the numerator indicates the power, or the number of times the number is to be taken as a factor, and the denominator indicates the root, or the number of equal factors into which that product is to be divided. Thus the square or second root of 12 is denoted by 121, the fourth root of by (?), and the square of the cube root of 27, or the cube root of the square of 27, is denoted by 271. 520. All the rational roots of whole numbers are also whole numbers, since every power of a fractional number is also a fractional number. 521. Prime numbers have no rational roots. A composite number, to have a given rational root, must have the exponent of the power of each of its prime factors exactly divisible by the exponent of that root. NOTE. — The number of composite numbers that have rational roots is comparatively small. The number of rational square roots of whole numbers from 1 to 250000 inclusive is only 500, and the number of rational cube roots of whole numbers from 1 to 8000000 inclusive is only 200. 522. The roots represented by the first ten numbers and their first six corresponding powers are shown in the following TABLE. 64 1st Power, 2d Power, 3d Power, 4th Power, 5th Power, 6th Power, 1 2 3 4 5 6 8 9 10 1 4 9 16 25 36 49 64 81 100 1 8 27 125 216 343 512 729 1000 1 16 81 256 625 1296 2401 4096 6561 10000 1 32 243 1024 3125 7776 16807 32768 59049 100000 1 64 729 4096 15625 46656 117649 262144 531441 1000000 NOTE. — It will be observed by the table, that a rational square root can only be obtained from numbers ending in 1, 4, 5, 6, or 9; or in an even number of ciphers, preceded by one of these figures. It is true, also, that, if the square number ends in 1, its square root ends in 1 or 9; if in 4, its square root ends in 2 or 8; if in 9, its square root ends in 3 or 7; if in 6, its square root ends in 4 or 6; and if in 5, its square root ends in 5. A perfect cube, however, may end in either of the nine digits, and in ciphers if the number of them is three or any multiple of three; also if the cube number ends in 1, its cube root will end in 1; if in 2, its cube root ends in 8; if in 3, its cube root ends in 7; if in 4, its cube root ends in 4; if in 5, its cube root ends in 5; if in 6, its cube root ends in 6; if in 7, its cube root ends in 3; if in 8, its cube root ends in 2; and if in 9, its cube root ends in 9. EXTRACTION OF THE SQUARE ROOT. 523. The extraction of the square root of a number is the process of finding one of its two equal factors; or of finding such a factor as, when multiplied by itself, will produce the given number. 524. The method generally adopted for extracting the square root depends upon the following principles : 1. The square of any number has, at most, only twice as many figures as its root, and, at least, only one less than twice as many. For the square of any number of a single figure consists of either. one or two places of figures, as 12 1, and 92 = 81; the square of any number of two figures consists of either three or four places, as 102 100, and 992 9801; and the same law holds in regard to numbers of three or more figures. Therefore, when the square number consists of one or two figures, its root will consist of one figure; when of three or four figures, its root will consist of two figures; when of five or six figures, its root will consist of three figures; and so on. Hence, if a number be separated into as many periods as possible of two figures each, commencing at the right, to these periods respectively will correspond the units, tens, hundreds, &c. of the square root of the number. 2. The square of a number consisting of TENs and units is equal to the square of the tens, plus twice the product of the tens into the units, plus the square of the units. Thus, if the tens of a number be denoted by a and the units by b, the square of the number will be denoted by (a + b)2 = a + 2 a b + 6?. Then, by this formula, if a = : 3, and 6 6, we have 3 tens + 6 units 30 + 6 36; and 362 = (30 + 6) = 302 + 2 X (30 X 6) + 62 = 1296. Or, analytically, 30 + 6 30+ 6 36 30 + 6 30+ 6 36 (a+b)x a = 302+30X6 900+180 1080 (a+b) x 6 30X6+62 180+36 216 (a+b) 302+2X (30X6)+62 = 900+360+36 = 1296 ato atb |