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Note. When two, or more powers are multi
plied together, their product is that power, whose index is the sum of the indices of the factors, or powers multiplied. * Evolution is the method of extracting any required root from any given power. Any number may be considered as a power of some other number; and the required root of any given power is that number, which, being multiplied into itself a particular number of times, produces the given power; thus, if 81 be the given
number, or power, its square, or second root is 9;
because 9 × 9 = 9*=81; and 3 is its biquadrate, or fourth root, because 3x3x3x3=3°=81. Again, if 729 be the given power, and its cube root be required, the answer is 9, for 9x9x9–729; and if the sixth root of that number be required, it is found to be 3, for 3x3x3x3x3x3=729. The required power of any given number, or root, can always be obtained exactly, by multiplying the number continually into itself; but there are many numbers, from which a proposed
root can never be completely extracted;—yet by
approximating with decimals, these roots may be found as exact as necessity requires. The roots that are found complete, are denominated rational roots, and those, which cannot be found complet
ed, or which only approximate, are called surd,
or irrational roots. Roots are usually represented by these characters or exponents;
y, or # which signifies the square root; thus, y 9, or 9% = 3
Likewise 83 signifies the square root of 8 cubed; and, in general, the fractional indices imply, that the given numbers are to be raised to such powers as are denoted by their numerators, and that such roots are to be extracted from these powers, as are denoted by their denominators.
Separate the given number into periods of two figures, by putting a point over the place of units,
* The reason for separating the figures in the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places ; the square of a number of two figures, of not more than four places, &c. So that there will be as many figures in the root, as the given number has periods so divided.
And, the reason of the several steps in the operation appears by assuming any root and raising it actually to the second power, as in common Arithmetic ; or from the Algebraic form of the square of any number of terms, whether two, three, or more. Thus,
Let 24 be squared, whose square is 576, put a for the ten’s place, and b for the units, a 4-b=24, therefore a2+2 a b-Hö2=576. The square root of the algebraic form is performed thus, - * ,
1st. Divisor a ) o: +2 a b + b2 (a + b
First, I find the root (a) of the first term (4%), and place it in the quo. tient, and its square under the first term, then I bring down the next term (2 ab) which is equal to twice the quotient (a) x by the next root (b) to be found, hence divide by twice the root (a) already found and it gives the next root, (b) which place in the quotient, as also to the right hand of the divisor, then I multiply both by (b) the reot last found, and subtract said product from the dividend; that is, (2 ab -- b2) which is equal to it, hence, the method of extracting the square root by the Algebraic form; but to apply this to the Arithmetical form, which is thus,
I find the nearest root to the first period which is 2, then I place this in the quotient and its square under the first period as in the Algebraic form, then I bring down the next period to the remainder, this dividend is equal to double the root found multiplied by the next figure of the root, plus the square of the figure to be found; hence I double the quotient figure for a divisor and set the figure found in the quotient, as also to the right hand side of the divisor, (because the first root found is in the ten's place) the same as in the Algebraic form, in the same manner it can be demonstrated if there be three or more terms,
another over the place of hundreds, and so on, over every second figure, both toward the left hand in whole numbers, and toward the right hand in the Decimal places.—When the number of integral places is odd, the first, or left hand period, will consist of one figure only. Find the greatest square in the first period on the left hand, and write its root on the right hand of the given number, in 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 divi, sor, and find how often it is contained in the said . dividend, exclusive of its right hand figure, and set this quotient both in the place of the quotient and in the divisor.—The best way of doubling the root, to form each new divisor, is to add the last figure always to the last divisor, as it is done in the subsequent examples. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it" the next period of the given number for a new dividend. Repeat the same operation again; that is, 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. f
Note 1. After the figures belonging to the given number are all exhausted, the operation may be continued in dicinals, by annexing any number of periods or ciphers to the remainder.
2. The number of integral places in the root
is always equal to the number of periods in the integral part of the resolvend. 3. When vulgar fractions occur in the given power, or number, they may be reduced to decimals, then the operation will be the same as before dictated.
Required the square root of 2981 16. Ans 546. Required the square root of 348.17320836. Ans. 18.6594. - * Required the square root of 17.3056. Ans. 4.16. Required the square root of .000729. Ans. .027. Required the square root of 17*. Ans. 4.168333+
To ExTRACT THE CUBE ROOT.
1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and also over every third figure, from thence to the left hand in whole numbers, and to the right in decimals. * 2. Find the greatest cube in the left hand period, and put its root in the quotient. 3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, and call this the resolvend. . 4. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor. 5. Seek how often the divisor may be had in the resolvend, and place the result in the quotient. *6. Multiply the triple square by the last quotient figure, and note the product; Multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under both, set the cube of the last quotient fi'gure, and call their sum the subtrahend.
* Among all the Rules that have been given by different Authors, I find that youth, (for whom this is chiefly designed,) can understand the Method of extracting the cube root by this with greater facility than any other Rule.