METHOD BY FACTORING. 635. The cube root of a number which is a perfect cube, may be found by resolving the given number into its prime factors, and taking the continued product of one third of each of those factors. EXAMPLE. 1. What is the cube root of 250047? SOLUTION.-The prime factors of this number are 3, 3, 3, 3, 3, 3, 7, 7, 7; therefore, 3 × 3 × 763, the required root. CUBE ROOT OF FRACTIONS. 636. The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. NOTES.-1. When one of the terms is a surd, as in this case, the fraction may be reduced to a decimal, and the root extracted to any required degree of exactness. 2. When the denominator is rational, the approximate root of the numerator may be found, and divided by the exact root of the denominator. APPROXIMATE CUBE ROOT OF FRACTIONS. 637. To extract the Cube Root of a common fraction which is not a perfect cube, to within less than its fractional unit. RULE.--Multiply the numerator by the square of the denominator, and extract the cube root of the product to within less than 1; divide this result by the denominator, and the quotient will be the required root. NOTE. This is equivalent to multiplying both terms of the fraction by the square of the denominator, and extracting the root of both terms. The denominator is thus made rational. EXAMPLES. 1. Find the cube root of to within less than SOLUTION. 7 x 16" 1792 and the cube root of 1792, to within less = than 1, is 12; therefore 1, or 2, is the required root. 2. Find the cube root of to within less than 1. 3. Required the cube root of to within less than 638. To extract the cube root of a common fraction which is not a perfect cube, to within less than any fractional unit. RULE.-Reduce the fraction to the same denomination as the given fractional unit, and then proceed according to the previous rule. EXAMPLES. 1. Required the cube root of 24 to within less than SOLUTION. 24 = 22 = and the cube root of 66, to within less than 1, is 4, and the cube root of 27 is 3; hence, or 13, is the required root. 2. Find the cube root of .8 within .01. 3. Find the cube root of .65 within .1. 4. Required the approximate cube root of ROOTS OF All Degrees. Ans. .92. Ans.. 639. Roots of higher degrees than square and cube roots may be extracted by the rules already given, when their indices contain no other factors than 2 or 3. Thus, the 6th root of a number is the square root of its cube root, or the cube root of its square root; for 6 = 2 x 3. The 9th root is the cube root of the cube root. The 12th root is the cube root of the square root of the square root of the given number; and so on. If the index of the required root is a prime number, as 5, 7, 11, etc., or contains any other factor, than 2 or 3, the following rule may be employed: RULE. For extracting any root whatever. I. Separate the given number into periods of as many figures each as there are units in the index of the required root. II. Find the greatest power in the left-hand period whose exponent corre sponds with the index of the required root, and place its root as the first figure of the required root; subtract the power from the left-hand period, and to the remainder bring down the next period for a dividend. III. Then, for a trial divisor, raise the part of the root already found to the power next inferior to the given power, and multiply the result by the index of the required root. IV. Find how many times the trial divisor is contained in the dividend, exclusive of all the figures last brought down, except the first, and write the quotient as the next figure of the root. V. Then, raise the two figures of the root found to the power whose exponent corresponds with the index of the required root, and subtract this power from the two left-hand periods of the whole given number, (unless it be found too large, in which case reduce the last root figure, and repeat the process till the right number is obtained,) and to the remainder bring down the next period for a dividend. VI. Proceed in the same manner to find the next figure of the root: then subtract the power of the three figures found from the first three periods of the given number: and so continue till the required root is obtained. NOTE.-There are several shorter methods of extracting roots of all degrees, deduced from algebraic formulas, but they are not of sufficient importance to students of arithmetic in general to demand attention in this connection. (See Strong's Algebra, p. 313; Hutton's Mathematics, p. 64.) 640. The method by the use of logarithms is, however, superior to all others. Persons having tables of logarithms at command, and understanding their use, can apply the following RULE. To extract any root of a number:— Find from a table the logarithm of the given number, and divide it by the index of the root: find from the table the number corresponding to the logarithm, and it will be the root required. NOTE. For practical applications of Square Root and Cube Root, see Mensuration. PRINCIPLES AND PRACTICAL APPLICATIONS OF MECHANICAL POWERS. 641. Mechanical Powers are certain simple machines, or elements of machinery, which convert a small force acting through a great space into a great force acting through a small space, or vice versa. They are usually considered to be six; namely, the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw. But, properly, two of these comprise the whole, -the Wheel and Axle and the Pulley being only modifications of the lever, and the Screw and the Wedge, of the inclined plane. 642. All of these machines act on the fundamental principle known as virtual velocities. According to this principle, the pressure or resistance is inversely as the velocities, or spaces passed through, or that would be passed through, if the piece were put in motion. THE LEVER. 643. The Lever in its simplest form is a bar of metal, wood, or other substance, capable of turning on a support, called the fulcrum. The mass to be raised, or resistance to be overcome, is called the Weight. The moving force applied at the other end of the bar is called the Power. The parts of the bar between the fulcrum and the points where the power and weight act are called the Arms of the lever. 644. The Principle or Law by which the lever acts as a mechanical power is that The power and resistance, or weight, are to each other inversely as their distances from the fulcrum, or centre of oscillation. This is the relation between the power and weight, without reference to friction, when they balance each other. If the weight is to be raised, additional power must be applied to overcome friction, and give it motion. |