23653,-the remaining part of the root. Hence we derive the following RULE. Separate the number into periods of three figures, by placing a point over the units' figure, and one over each third figure to the left, (and also to the right, if decimals are desired in the root). Write in the quotient the root of the greatest cube contained in the left hand period, and subtract the cube from the period, annexing the second period to the remainder. Take three times the square of the root already found, for a trial divisor, and find how many times this divisor is contained in the hundreds of the dividend. Place the result in the quotient, and its square at the right of the trial divisor (supplying the place of tens with a zero, if the square is less than ten). Multiply 30 times the root already found, by the last root figure; this product added to the divisor, will give the complete divisor. Multiply the complete divisor by the last root figure, subtract the product from the dividend, annex the next period to the remainder, and proceed as before until the whole root is extracted. If any complete divisor is not contained in its dividend, a zero must be annexed to the root, and two to the trial divisor, and the next period brought down for a new dividend. If any figure obtained for the root is too great, diminish it by 1 or more, and repeat the work. [It may be well to illustrate to the pupil the algebraical formation of the cube, by raising 30+7 to the third power. In forming the complete divisor, we say, "Multiply 30 times the root already formed, by the last root figure," because 3× the tens X the units would give tens, and a zero must be annexed for units. The square of the units is written at the right of the trial divisor, to render the work more compact. This square being units, and the trial divisor hundreds, the two may very properly be united.] The trial divisors, after the first, may be more conveniently found, by adding to the last complete divisor the last number used to complete it, and twice the square of the last root figure. Thus in the foregoing example, our third trial divisor= 1096281 trial divisor, 1112643 1. What is the cube root of 328509? 2. What is the cube root of 1986091 ? 3. What is the cube root of 9129329? 4. What is the cube root of 345.357380096? 27 13 7. What is the cube root of 53; 1138; 2748? 9. What is the cube root of 148877? 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 11 oz., what would be the value of a gold ball 1 foot in diameter ? 18. What is the size of á ball that weighs 27 times as much as one 3 feet 6 inches in diameter ? 19. If a hollow sphere 3 feet in diameter 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 divisible 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=3×2× 2, 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 FOR EXTRACTING THE ROOTS OF ALL POWERS. 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 column 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 number 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. 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 assistance, but we are obliged to rely upon trial, for the first figure of the root, which being ascertained, the succeeding figures will be easily found.] |