9. What is the cube root of 146877 ? 1030301 10. What is the cube root of .0081 ? 11. What is the cube root of 4.160075243787 ? The solid contents of similar bodies are to each other as the cubes of their diameters, or of their similar sides. The solid contents of a sphere may be found by multiplying the cube of the diameter by .5236. 12. What are the solid contents of the earth, supposing it a perfect sphere, whose diameter is 7920 miles ? 13. If a ball 2 inches in diameter, weighs 1 ; pounds, what would be the weight of a similar ball 6 inches in diameter ? 14. What is the side of a cubical box that will hold 1 bushel ? 15. What is the side of a cubical pile that contains 258 cords of wood ? 16. If a tree 1 foot in diameter, yields 2 cords of wood, how much wood is there in a similar tree that is 3ft. 6in. in diameter ? 17. If a pound avoirdupois of gold is worth $200, and a cubic inch weighs 1140%., what would be the value of a gold ball 1 foot in diameter ? 18. What is the size of a ball that weighs 27 times as much as one 3 feet 6 inches in diameter ? 19. If a hollow sphere 3 feet in diarneter and 23 inches thick, weigh 12 tons, what would be the dimensions of a similar sphere that would weigh 324 tons ? 20. What is the side of a cubical block of wood that weighs as much as a sphere 15 inches in diameter ? PROPERTIES OF CUBES. If a cube be divisible by 6, its root will also be divisi. ble by 6. And if a cube, when divided by 6, has any remainder, its root divided by 6 will have the same remainder. All exact cubes are divisible by 4, or can be made so by adding or subtracting 1. All exact cubes are divisible by 7, or can be made so by adding or subtracting 1. All exact cubes are divisible by 9, or can be made so by adding or subtracting 1. Every cube is divisible by the cube of each of its prime factors. ROOTS OF HIGHER POWERS. When the exponent of a power can be resolved into two or more factors, by successively extracting the roots denoted by those factors, we may obtain the root desired. Thus, as 12=3x2x2, the cube root of the square root of the square root of a number, is equal to the 12th root. So the square root of the square root=the 4th root; the cube root of the cube root is the 9th root; the cube root of the square root is the 6th root, &c. The demonstration of the following rule depends upon Algebraical principles, and therefore cannot properly be introduced here. In its application to square and cube roots, the student will be able to trace some analogy to the rules already given. GENERAL RULE At the left of the number whose root is required, arrange as many columns as are equal to the index of the root, writing 1 at the head of the first or left hand column, and zero at the head of each of the others. Divide the number into periods of as many figures as the index of the root requires. Write the root of the left hand period as the first figure of the true root. Multiply the number in the first column by the root figure, and add the product to the second column; add the product of this sum by the root figure to the third column, and so proceed, subtracting the product of the last coJurn from the given number. Repeat this process, stopping at the last column, and thus proceed, stopping one column sooner each time, until the last sum falls in the second column. To determine the second root figure, consider the num. ber in the last column as a trial divisor, and proceed with the second root figure thus obtained, precisely as with the first. Continue the operation until the root is completed, or the approximation carried as far as is desired. In order to avoid error, observe carefully the value of each root figure and each product. Thus, if the first root figure is hundreds, the number in the second column will be hundreds,-in the third, ten thousands,—in the fourth, millions, &c. EXAMPLES FOR THE BOARD. What is the third root of 205692449327 ? 205692449321(5903 125 bill. (2) 17703 un. 104483109 c. d. The complete divisors are marked c. d., the trial divisors, t. d. The figures at which the new additions commence, are marked (1), (2). The partial dividends by which each root figure is determined, are distinguished by a comma. They always terminate with the first figure of the period that is annexed. The abbreviations, thous., mill., &c., show the value of the figures against which they are placed. [The extraction of the square root, and the solution of equations of the second power (of which examples are given on p. 125), afford very ready and convenient applications of this rule. The determination of the first root figure in the higher powers, would frequently be difficult, without the aid of Table V. In the solution of many Algebraical Equations, even this table affords no assist. ance, but we are obliged to rely upon trial, for the first figure of the root, which being ascertained, the succeed. ing figures will be easily found.] Extract the 5th root of 858533232.56832. 0 0 858533232.56832(612 6 teng. 36 hund. 216 thous. 1296 ten thou. 7776 hund, thous. 36906 304 304 1 (2) 37210 6104 hund. (2) 3052 tenths 3727104 The additions to the left hand column may be made mentally, and thus shorten the labor. There are other abbreviations, for which the student is referred to the Chapter on Numerical Approximations. The first root figure in each of the following examples, may be found by the table of Powers and Roots. 1. Extract the square root of 350026681. 2. Extract the square root of 3; 5; 6.5. 3. Extract the cube root of 2924207. 4. Extract the cube root of 13; 12.5. 5. Extract the fifth root of 65.7748550151. 6. Extract the 7th root of 1.246688292353624506368. To show the universal application of the rule, we will solve a question containing many of the powers of a num. ber. At the head of the columns we write the coefficients* of all the powers, from the highest to the lowest, substituting a 0 when any power is wanting. The first coefficient is then multiplied by the root figure, and the product ad. * A coefficient is a figure indicating the number of times any term is employed. Thus, in 7 times the 5th power, 7 is the coefficient of the 3th power. ded to the second coefficient, the product of this sum by the root figure, added to the third coefficient, and so on. EXAMPLE FOR THE BOARD. 7 times the 5th power, minus 2 times the 3d power, plus 5 times the second power of a certain number, is equal to 1405569. 53125. What is the number ?* We commence with writing in order the coefficients, 7 for the 5th power, 0 for the 4th power,-2 for the 3d power, 5 for the 2d power, and 0 for the first power. Then, as the number has two integral periods, the first figure of the root will be tens. Finding by trial that 1 is the first root figure, we add 10x7 to 0; 10 X 70 to -2; &c., as in the following solution: 7 0 1405569.53123(11.5 700 6980 69850 698500 -2 5 70 388.5 * The equation stated algebraically, would be, 725 - 2x3 + 5.38 = 1405569.53125. |