When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multiplied. Or, the multiplication of the powers answers to the addition of the indices. Thus, in the following powers of 2. The 2d power of 45 is 2025. The square of 4·16 is 17.3056. The 5th power of 029 is 000000020511149. The square of is . 729 The 3d power of is 25. EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. = The root of any number, or power, is such a number as, being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 22 2 × 2 4; and 3 is the cube root or 3d root of 27, because 33 3 × 3 × 3 = 27. = = Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals we may approximate or approach towards the root to any degree of exactness. These roots, which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also, the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd, or irrational. Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root against it. Thus, the third root of 20 is expressed by 20; and the square root or 2d root of it is 20, the index 2 being always omitted when the square root is designed. When the power is 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 third root of 45 12 is 45 — 12, or thus, (45 12), enclosing the numbers in parentheses. But all roots are now often designed like powers, with fractional indices: thus, the square root of 8 is 8, the cube root of 25 is 25*, 18+, or, (45 — 18). and the 4th root of 45 18 is 45 TO EXTRACT THE SQUARE ROOT. RULE. Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on, over every second figure, both to the left hand in integers, and to the right in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in Division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period for a dividend. Double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right-hand figure; and set that quotient figure both in the quotient and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to the next period of the given number, for a new dividend. Repeat the same process over again, namely, find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before, and so on through all the periods to the last. The best way of doubling the root to form the new divisor is by adding the last figure always to the last divisor, as appears in the following examples. Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period. To find the square root of 29506624. When the root is to be extracted to many places of figures, the work may be considerably shortened, thus: Having proceeded in the extraction after the common method till there be found half the required number of figures in the root, or one figure more; then, for the rest, divide the last remainder by its corresponding divisor, after the manner of the third contraction in Division of Decimals; thus, To find the root of 2 to nine places of figures. 2(1.4142 The square root of 1.41421356 the root required. 000729 is 027. The square root of 3 is 1.732050. The square root of 5 is 2.236068. The square root of 6 is 2-449489. RULES FOR THE SQUARE ROOTS OF COMMON FRACTIONS AND MIXED NUMBERS. First, prepare all common fractions by reducing them to their least terms, both for this and all other roots. Then, 1. Take the root of the numerator and of the denominator for the respective terms of the root required. And this is the best way if the denominator be a complete power; but if it be not, then, 2. Multiply the numerator and denominator together; take the root of the product: this root being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the fractional root required. And this rule will serve whether the root be finite or infinite. 3. Or reduce the common fraction to a decimal, and extract its root. 4. Mixed numbers may be either reduced to improper fractions, and extracted by the first or second rule; or the common fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. By means of the square root, also, may readily be found the 4th root, or the 8th root, or the 16th root, &c.; that is, the root of any power whose index is some power of the number 2; namely, by extracting so often the square root as is denoted by that power of 2; that is, two extractions for the 4th root, three for the 8th root, and so on. So, to find the 4th root of the number 21035-8, extract the square root twice as follows: 1. DIVIDE the page into three columns (1), (II), (III), in order, from left to right, so that the breadth of the columns may increase in the same order. In column (III) write the given number, and divide it into periods of three figures each, by putting a point over the place of units, and also over every third figure, from thence to the left in whole numbers, and to the right in decimals. 2. Find the nearest less cube number to the first or left-hand period; set its root in column (III), separating it from the right of the given number by a curve line, and also in column (1); then multiply the number in (1) by the root figure, thus giving the square of the first root figure, and write the result in (II); multiply the number in (II) by the root figure, thus giving the cube of the first root figure, and write the result below the first or left-hand period in (III); subtract it therefrom, and annex the next period to the remainder for a dividend. 3. In (1) write the root figure below the former, and multiply the sum of these by the root figure; place the product in (11), and add the two numbers together for a trial divisor. Again, write the root figure in (1), and add it to the former sum. 4. With the number in (II) as a trial divisor of the dividend, omitting the two figures to the right of it, find the next figure of the root, and annex it to the former, and also to the number in (1). Multiply the number now in (1) by the new figure of the root, and write the product as it arises in (11), but extended two places of figures more to the right, and the sum of these two numbers will be the corrected divisor; then multiply the corrected divisor by the last root figure, placing the product as it arises below the dividend; subtract it therefrom, annex another period, and proceed precisely as described in (3), for correcting the columns (1) and (11). Then with the new trial divisor in (II), and the new dividend in (III), proceed as before. When the trial divisor is not contained in the dividend, after two figures are omitted on the right, the next root figure is 0, and therefore one cipher must be annexed to the number in (1); two ciphers to the number in (II); and another period to the dividend in (III). When the root is interminable, we may contract the work very considerably, after obtaining a few figures in the decimal part of the root, if we omit to annex another period to the remainder in (III); cut off one figure from the right of (II), and two figures from (1), which will evidently have the effect of cutting off three figures from each column; and then work with the numbers on the left, as in contracted multiplication and division of decimals. Find the cube root of 21035.8 to ten places of decimals. (11) 4 8 (III) 21035-8 (27-60491055944 7 74 12.. 469 1669 518 2187 8 13035 11683 1352800 11224... Required the cube roots of the following numbers: 48228544, 46656, and 15069223. 64481.201, and 28991029248. 12821119155125, and 000076765625. 18824 and 16. 428759 91, and 7%. 364, 36, and 247. 40.1, and 3072. 23405, and 0425. 2, and 2.519842. 4.5, and 1.98802366. |