square, extracting therefore the integral part of the root of the numerator we Again, let it be required to extract the square root of 3.425. 3425 But 1000 is not a perfect square, it is This fraction is the same as however equal to 100 × 10, or (10)a × 10; thus, in order to render the denominator a perfect square it is sufficient to multiply both terms of the fraction by 10, which gives 34250 34250 or Extracting the integral part of the root 185 100' 34250 we find 185, hence the root required is or 1.85, a result which is If we wish to have a greater number of decimal places in the root, we must add on the right of 34250 twice as many zeros as we wish to have additional de cimal figures. From what has just been observed, we readily deduce the general rule for the extraction of the square root of a decimal fraction which has been already given in our Arithmetic. EXTRACTION OF THE CUBE ROOT OF NUMBERS. 91. The numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, when cubed become 1000, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1000000, 1000000000; and reciprocally, the numbers in the first line are the cube roots of the numbers in the second. Upon inspecting the two lines we perceive, that, among the numbers expressed by one, two, or three figures, there are only nine which are perfect cubes, consequently, the cube root of all the rest must be a whole number plus a fraction. 92. But we can prove, in the same manner as in the case of the square root, that the cube root of a whole number, which is not the perfect cube of some other whole number, cannot be expressed by an exact fraction, and consequently its cube root is incommensurable with unity. a an exact fraction in its lowest terms, to be the cube root α a 3 For if we suppose ō' of some whole number N, it follows that the cube of or must be equal b' b39 to N. But since a and b are, by supposition, prime to each other, a3 and b3 3 are also prime to each other; and therefore cannot be equal to a whole number. 93. The difference between the cubes of two consecutive whole numbers is greater in proportion as the numbers themselves are greater; the expression for this difference can easily be found. Hence, (a+1)3 — a3 = 3a2+3a+1; that is to say, the difference of the cubes of two consecutive whole numbers is equal to three times the square of the less of the two numbers, plus three times the simple power of the number, plus unity. Thus, the difference between the cube of 90 and the cube of 89 is equal to 3x (89)2+3x89+1=24031. Let us now proceed to investigate a process for the extraction of the cube root of any number. EXTRACTION of the Cube Root. 94. The cube root of a proposed number, consisting of one, two, or three figures only, will be found immediately by inspecting the cubes of the first nine numbers in Art. 91. Thus the cube root of 125 is 5, and the cube root of 54 is 3 plus a fraction, for 3×3×3=27, and 4×4×4=64; therefore 3 is the approximate cube root of 54 within one unit of the true value. For the purpose of investigating a new and simple rule for the extraction of the cube root, it will be necessary to attend to the composition of a complete power of the third degree. Now, since we have (a+b)3=(a+b) (a+b) (a+b)=a3+3a2b+3ab2+b3, it is obvious that the cube of a number, consisting of tens and units, will be algebraically indicated by the polynomial a3+3a2b+3ab2+b3 where a designates the number of tens, and b the number of units in the root sought. The number in the tens' place will evidently be found by extracting the cube root of the monomial a3, for Va3-a, and removing a3 from the polynomial a3+3a2b+3ab2+b3, we have the remainder 3a2b+3ab2+b3=(3a2+3ab+b2) b; and the difficulty that has been hitherto experienced in the extraction of the cube root entirely consists in the composition of the expression 3a2+3ab+b2, which is obviously the true divisor for the determination of b, the figure of the root in the place of units. The part 3a2 of the expression 3a2+3ab+b2, being independent of b, the yet unknown part of the root, is employed as a trial divisor for the determination of b; but since the expression 3a2+3ab+b2 involves the unknown part of the root in its composition, it is obvious that the trial divisor 3a2, which does not contain b, will at the first step of the operation give no certain indication of the next figure of the root, unless the figure denoted by b be very small in comparison with that denoted by a; for the trial divisor 3a2 will be considerably augmented by the addend 3ab+b2, when b is a large number, while the augmentation, when b is a small number, will not so materially affect the trial divisor. When the figure in the tens' place is a small number, as 1 or 2, it is hence obvious that little or no dependence can be placed on the trial divisor; but if a be great and b small, the trial divisor 3a2 will generally point out the value of b. All this will be evident if we consider that the relative values of a and b materially affect the true divisor, 3a2+3ab+b2. In the successive steps, however, of the cube root, this uncertainty diminishes; for conceiving a to designate a number consisting of tens and hundreds, and b the number of units, then the value of b being small in comparison with a, the amount of the effect of b in the addend 3ab+b2 will be very inconsiderable; hence the trial divisor, 3a2, will generally indicate the next figure in the root. To remove, in some measure, the difficulty which has hitherto been experienced in the extraction of the cube root, we shall proceed to point out two methods of composing the true divisor, 3a2+3ab+b2, and leave the student to select that which he conceives to possess greater facility of operation. Distinguishing the three columns from left to right, by first, second, and third columns, we write a in the root, and also three times vertically in the first column; then a X a produces a2, which write also three times vertically in the second column; multiply the second a2 by a, placing the product, `a3, under a3 in the third column; then subtracting a3 from the proposed quantity, we have the remainder 3a2 b+3ab2+b3. The sum of the three quantities in the second column gives 3a2 for the trial divisor, by which find b, the next figure of the root, and to 3a, the sum of the three last written quantities in the first column, annex b; then the sum, 3a+b, is multiplied by b, and the product, Sab+b2, is placed in the second column; then the trial divisor 3a2, and the addend 3ab+b2 being collected, give the true divisor, 3a2+3ab+b2, which multiply by b, and place the product, 3a2 b+3a b2+b3, under the remainder 3a2 b+3ab2+b3. When there is a remainder after this operation, the process may be continued by writing b twice in the first column, under 3a+b, and b2 once in the second column, under the last true divisor; then 3a2+6ab+3b2, the sum of the last written three lines in the second column, will be another trial divisor, with which proceed as above. We have written a2 in the second column three times in succession, to assimilate the first step in the operation to the other successive steps, but the first trial divisor, 3a2, may be written at once, and the symmetry of the disposition of the quantities in the first steps disregarded 96. Second method of composing 3a2+3a b+b2, the true divisor. In this method we write a under a in the first column, and the sum 2a being multiplied by a, gives 2a2 to place under a2 in the second column, and the sum of 2a2 and a2 is 3a2 for the trial divisor. Again, under 2a in the first column write a, and the sum of 2a and a gives 3a. Now, having found b by the trial divisor, annex it to 3a in the first column, making 3a+b, which, multiplied by b, and the product placed in the second column, gives, by addition, the true divisor 3a2+3ab+b2, as before. We shall exhibit the operation of extracting the cube root by both these methods. EXAMPLES. (1.) What is the cube root of x-9x3+39x1—99x3+156x2 —144x+64 ? (2.) What is the cube root of x+6x3-40x3 +96x-64? (3.) What is the cube root of a3+3a2b+3a b2+b3+3a2 c+6a b c +3b2c +3ac2+36 c2+c? Ans. a+b+c. Ans. x2-2x+ (4.) Extract the cube root of xo—6x3+15x1—20x3+15x2-6x+1. 97. The same process is employed in the extraction of the cube root numbers, as in the subsequent examples. |