multiplication the root is used twice; once by being taken as the multiplicand, and once more as the multiplier. A fraction is involved by involving both its numerator and its de 516. To raise a number to any required higher power, with out producing all the intermediate powers. Ex. 1. What is the 7th power of 5? 1 2 OPERATION. Ans. 78125. + 2 + 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. 5. What is the 5th power of 195? Ans. 281950621875. 6. What is the 6th power of ? 8. Required the 2d power of 6031. 12. What is the 4th power of 4.367 ? Ans. 64 729. Ans. 36372961. Ans. 96889010407. Ans. 205891132094649. 13. Involve the following numbers to the powers denoted by their respective exponents: (22)5, 1.045, and (3)*. Ans. 157,288; 1.800943+; 1161225 2401 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. 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 is denoted by }. 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 12, the fourth root of by (2), and the square of the cube root of 27, or the cube root of the square of 27, is denoted by 273. 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 2d Power, 1 4 125 625 3125 216 343 512 729 1000 1296 2401 4096 6561 10000 7776 16807 32768 59049 100000 3d Power, 1 8 27 64 4th Power, 1 16 81 256 5th Power, 1 32 243 1024 6th Power, 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. 523. EXTRACTION OF THE SQUARE ROOT. 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 : = a2 + 2 a b + b2. Then, by this formula, if a = 3, and b +6 units = 30 362 = +6 36; and (30+6)2 Or, analytically, a+b a + b = = 6, we have 3 tens (a+b)x a 302+2X(30×6)+62 = 900+360+36 = It is evident, as evolution is the reverse of involution (Art. 517), that from the process now given of obtaining a square may be deduced a method of extracting its root. Since the square of (a+b) is a2 + 2 a b + b2, the square root of a2+2ab+b2 must be ab. Now it will be observed that a, the first term of the root, is the square root of a2, the first term of the square; and if a2 be subtracted, there will remain 2 ab+b2, from which b, the second term of the root, is to be obtained. But 2 a b + b2 is the same as (2 a + b) × b, therefore the remainder equals (2 a + b) × b. But as b, the units, is always much less than 2 a, twice the tens, we consider that 2 a Xb is about equal to the whole remainder, and taking 2a (which we know) as the trial divisor, we obtain b, the units. But as the true divisor is 2 ab, we add the units to twice the tens and multiply the sum by the units, which gives a product equal to the whole remainder, or 2 a b + b2. Since every number of more than one figure may be considered as composed of tens and units, we may have tens and units of units, tens and units of tens, tens and units of hundreds, &c. Hence, the principle just explained applies equally whether the root contains two or more than two figures. 525. To extract the square or second root of numbers. Ex. 1. What is the square root of 1296? OPERATION. 129636 9 66396 396 0 = Ans. 36. Beginning at the right, we separate the number into periods of two figures each, by placing a point () over the right-hand figure of each period. Since the number of periods is two, the root will consist of two figures, tens and units. Then 1296 the square of the tens plus twice the product of the tens into the units, plus the square of the units. The square of tens is hundreds, and must therefore be found in the hundreds of the number. The greatest number of tens whose square does not exceed 12 hundreds is 3, which we write as the tens figure of the root. We subtract the 9 hundreds, the square of the 3 tens, from the 12 hundreds, and there remain 3 hundreds; after which we write the figures of the next period, and the remainder is 396 twice the product of the tens into the units plus the square of the units. We have then next to find a number which, added to twice the 3 tens of the root, and multiplied into their sum, shall equal 396. By dividing this remainder |