Any number, whether given, or assumed at pleasure, may be called the root, or first power of this number; and its other powers are the products, that result from multiplying the number by itself, and the last product by the same number again; and so on to any number of multiplications. The index, or exponent, is the number denot ing the height, or degree of the power, being always greater by one, than the number of multiplications employed in producing the power. It is usually written above the root, as in the following EXAMPLE, where the method of involution is plainly exhibited. What is the second power of 3.05? Ans. 9.3025 What is the third power of 85.3? Answer, 620650.477 What is the fourth power of .073? Answer, 090028398241 What is the eighth power of .09? Answer, .00.00.00.0043046721 Note. When two, or more powers are multiplied 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 9×9-92-81; and 3 is its biquadrate, or fourth root, because 3×3×3×3=34-81. Again, if 729 be the given power, and its cube root be required, the answer is 9, for 9×9×9=729; and if the sixth root of that number be required, it is found to be 3, for 3×3×3×3×3×3=729. The required power of any given number, or root, can always be obtained exactly, by multiplying the number continually into it self; 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 completed, or which only approximate, are called surd, or irrational roots. Roots are usually represented by these characters or exponents; Likewise: 8 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. RULE For extracting the Square Root. Separate the given number into periods of two figures, by putting a point over the place of units, 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 divisor, 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. Note 1. After the figures belonging to the given number are all exhausted, the operation may be continued in decimals, by annexing any number of periods of 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. EXAMPLES. Required the square root of 1710864. 1 | 1710864(1308,- Answer. 1 23 | 71 260820864 Required the square root of 298116. Ans. 546. Required the square root of 348.17320836, 18.6594. Ans. 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+ A GENERAL RULE For extracting any Root whatever. Find by trial a number, which, when involved to the power denoted by the index of the required root, shall come nearest to the given number, whether greater or less; and let that number be called the assumed root, and when thus involved, the assumed power, |