(6), the left hand figure of the fourth period is annexed, and (3056), the resulting quantity, divided by (436), twice (218), the ROOT last found, gives (7), the fourth and last figure of the root; and this (7), placed on the right of (218), the ROOT last found, gives (2187), the SQUARE ROOT of (4782969), the GIVEN NUMBER, the square of which exactly coincides with the number itself, and therefore, the work is right. In the same way we may extract the CUBE ROOT required in the second example, as follows: First. Because it is the THIRD, or CUBE ROOT, which is demanded, the number, (387420489), from which that root must be extracted, is regarded as a cube, or third power, and must, therefore, be separated into PERIODS of three figures, EACH. Thus : given cube. cube root. 387420489(729. ANSWER. 73-343, 1st subtrahend. (72)x3=147)444(2, 3d figure in the root. 1st and 2d periods, 387420, 2d minuend. 723-373248, 2d subtrahend. (722)x3=15552)141724(9, 3d figure in the root. given cube, 387420489, 3d minuend. 7293387420489, 3d subtrahend. Then, second, the nearest CUBE ROOT of 387 is 7, the cube of which (7), is 343, (1st subtrahend,) which, taken from 387, (the period which produced it,) leaves 44, to which annex 4, and we have 444, (first dividend,) which, divided by 3 times the square of 7 to 72 × 3 = to 147, gives 2 for the second figure in the root, which, placed on the right of the first, gives 72 for the cube root of 387420, (the periods which produced it.) Now, if we cube 72, we shall have 373248, which, taken from 387420, leaves 14172, to the right of which we will annex 4, (the left hand figure of the third and last period,) making 141724, (the second dividend,) which, divided by (3 times the square of 72 to 722 × 3 = to 5184 x 3: = to) 15552, gives 9 for the third and last figure in the root, which, annexed to 72, gives 729, for the root sought, the cube of which is exactly equal to (387420489), the number whose cube root was demanded, and therefore, the work is right. In the same manner may be extracted the fifth, seventh, eleventh, and in general, all other ROOTS, whose indices (plural of index,) are prime numbers, and by the aid of these any other roots whatever, may be found; thus, the fourth root may be found by extracting the square root twice; the sixth root may be found by first extracting the CUBE, and then the square root; the eighth root, by first extracting the square root, which will give the fourth power, of which extract the square root again, and it gives the square, of which extract the square root a third time, and the result will be the eighth root, &c. If the previous examples are studied carefully, the pupil will be prepared to adopt the following General RULE for Extracting all Roots. First. Separate the given number (from right to left,) into periods, containing as many figures in each as the index of the REQUIRED ROOT, contains units, viz. two figures for the square root; three figures for the cube (third,) root, &c. Second. Find by trial, a number which being raised to a power corresponding with the index of the required root, will come nearest to, but less than, the first (left hand) period, this NUMBER will be the FIRST FIGURE in the REQUIRED ROOT which, like a quotient, we place on the right of the NUMBER whose ROOT we are extracting. Third. Raise the ROOT now found to a power whose index is that of the REQUIRED ROOT, then subtract this power from the period, or periods used, and annex the left hand figure of the next period to the REMAINDER for a DIVIDEND. Fourth. Raise the ROOT last found, to a power whose index is one less than the index of the REQUIRED ROOT, and multiply this power by the latter index the PRODUCT will be the DIVISOR. Fifth. Divide the DIVIDEND by the DIVISOR, and the quoTIENT will be the NEXT FIGURE in the REQUIRED ROOT, which (placed on the right of the FORMER,) will give the REQUIRED ROOT of the periods used, with WHICH proceed as directed in sections third, fourth, and fifth; this process con tinued till all the periods are used, will give the ROOT SOUGHT, or ANSWER. NOTE I.-If the quantity whose ROOT is required, is a vulgar fraction, extract the root of the numerator and denominator separately, the resulting FRACTION will be the ANSWER. NOTE II.-If the QUANTITY is a mixed number, reduce IT to an improper fraction or if IT consists of an integer and a decimal, separate IT into periods, both right and left, from the point, and remember, when the first period in the decimal has been brought down, the next figure in the root will be TENTHS, after which the process may be continued at pleasure, till the ROOT is sufficiently near. NOTE III.-The index of a ROOT is the same as the index of a POWER; but being the reverse in signification, it is sometimes placed as a denominator of the latter; thus, 63 signifies that the SQUARE of 6 must have the CUBE ROOT extracted; but more commonly only thus, /6to 36 to 3.301927; also, 3/27=3, or 275=3; 3√125=5, or 125=5, &c. This character (√) is called the the radical (root) sign, and when placed before a number without an index, it signifies that the square root of that number must be extracted; thus, √9=3; but when any other than the square root is required, it must have an index; as, 3/27=3, &c. NOTE IV. When the root of a number cannot be found exactly, that number is called a surd, to extract the root of which, we generally find, first, the whole number which comes nearest the REQUIRED ROOT, and then affix ciphers to the remainder, sufficient to carry the DECIMAL in the root to three places, except when great accuracy is required, and then IT may be carried to as many places as may be thought necessary. For the sake of further illustrating the correctness and general application of the RULE, we will give a few additional EXAMPLES. 1. What is the square root of 899? Here the REQUIRED ROOT is the square root, and therefore, the index is 2, which will enable us to proceed according to the RULE; hence, we must allow two figures to each period, and since a portion of the POWER is a decimal, we must separate the periods according to Note II., after which we observe the RULE in every particular, the same as if the whole power were an integer, observing to place a decimal point in the root, when we bring down the first decimal figure. Thus: power. sq. root. 899.000000(29.983+. ANSWER. 224, 1st subtrahend. 2x2=4)49(9, 2d figure in the root. 292-841, 2nd subtrahend. 2992-89401, 3d subtrahend. 8990000, 4th minuend. 299828988004, 4th subtrahend. 29982=5996)19960(3, 5th figure of root. 899000000, 5th minuend. 299832-898980289, 5th subtrahend. 19711 remains. 2. What is the cube root of 849278.123? given power. root. Thus : 849278.123(94.7. Answer. 93 729, 1st subtrahend. 92x3=243)1202(4, 2d figure in root. 849278, 2d minuend. 943 830584, 2d subtrahend. 942+3=26508)196941(7, 3d figure in root. 849278123, 3d minuend. Here we are required to extract the CUBE ROOT, the index of which is three. The given power, or number, (849278.123), consists of a whole number, and a decimal; and therefore, first, beginning at the point, we find the whole number (849278), contains two PERIODS, and the decimal (.123), contains one PERIOD; second, the CUBE ROOT of the INTEGER (849278), is 94, with a remainder of 19694, to the right of which we annex 1, (the left hand figure of (.123), the decimal period,) which gives 196941, (second dividend,) and this contains 26508 (942 × 3) 7 times, which (7), is the first decimal figure, or the third and last figure in the required root, 94.7, which is the ANSWER sought. NOTE. The biquadrate, or fourth root of a given number, or power, is found by first extracting the square root of that power, which reduces it to a square, and then by extracting the square root of that square, which gives the PRIME ROOT of the GIVEN NUMBER, or FOURTH ROOT Sought. On the same principle, we may find the sixth root, by first extracting the CUBE ROOT, which reduces the index to 2, and then by extracting the sQUARE ROOT, which reduces the index to 1, which gives the PRIME, or SIXTH ROOT sought, &c. 3. What is the biquadrate, or fourth root of 1677.7216? |