first term (a3), and the next term is 3a2b. Three times the square of the first letter or term of the root multiplied by the 2d term of the root. Therefore, to find this second term of the root, we must divide the second term of the power (3a2b) by three times the square of the root already found (a). 3a2)3a2b(b 3a2b When we can decide the value of b, we may obtain the complete divisor for the remainder, after the cube of the first term is subtracted, thus: Take out the factor b, and 3a2+3ab+b2 is the complete divisor for the remainder. But this divisor contains b, the very term we wish to find by means of the divisor; hence, it must be found before the divisor can be completed. In distinct algebraic quantities there can be no difficulty, as the terms stand separate, and we find b by dividing simply 3ab by 3a2; but in numbers the terms are mingled together, and b can only be found by trial. Again, the terms 3a2+3ab+b2 explain the common arithmetical rule, as 3a2 stands in the place of hundreds, it corresponds with the words: "Multiply the square of the quotient by 300," "and the quotient by 30," (3a), &c. By inspecting the various powers of a+b (Art. 73), we draw the following general rule for the extraction of roots: RULE. Arrange the terms according to the powers of some letter; take the required root of the first term and place it in the quotient; subtract its corresponding power from the first term, and bring down the second term for a dividend. Divide this term by twice the root already found for the SQUARE root, three times the square of it for the CUBE root, four times the third power for the fourth root, &c., and the quotient will be the next term of the root. Involve the whole of the root thus found, to its proper power, which subtract from the given quantity, and divide the first term of the remainder by the same divisor as before; proceed in this manner till the whole root is determined. EXAMPLES. 1. What is the cube root of 2+6x-40x+96x-64? 2+6x-4023+96x-64 (x2+2x-4 Divisor 3x1) 6x3=1st remainder. 2+6x+12x+8x=(x2+2x)3 Divisor 324 )-12-2d remainder. x+6x5—40x3+96x-64 2. What is the cube root of 27a+108a2+144a+64? Ans. 3a+4. 3. What is the cube root of a3-6a2x-12ax2-8x3 ? Ans. a-2x. 4. What is the cube root of 26-3x2+5x3-3x-1 ? Ans. x2-x-1. 5. What is the cube root of a3—6a2b+2ab2—872 ? Ans. a-2b. 3 1 6. What is the cube root of 3+3x+-+? 1 Ans. x+ (ART. 81.) To apply this general rule to the extraction of the cube root of numbers, we must first observe that the cube of 10 is 1000, of 100 is 1000000, &c.; ten times the root producing 1000 times the power, or one cipher in the root producing 3 in the power; hence, any cube within 3 places of figures can have only one in its root, any cube within 6 places can have only two places in its root, &c. Therefore, we must divide off the given power into periods consisting of three places, commencing at the unit. If the power contains decimals, commence at the unit place, and count three places each way, and the number of periods will indicate the number of figures in the root. Here, 12 is contained in 48, 4 times; but it must be remembered that 12 is only a trial or partial divisor; when completed it will exceed 12, and of course the next figure of the root cannot exceed 3. The first figure in the root was 2. Then we assumed a=2. Afterward we found the next figure must be 3. Then we assumed a=23. To have found a succeeding figure, had there been a remainder, we should have assumed a=234, &c., and from it obtained a new partial divisor. 2. What is the cube root of 148877? Ans. 53. Ans. 83. Ans. 111. Ans. 127. (ART. 82.) The methods of direct extraction of the cube root of such numbers as have surd roots, are all too tedious to be much used, and several eminent mathematicians have given more brief and practical methods of approximation. One of the most useful methods may be investigated as follows: Suppose a and a+c two cube roots, c being very small in relation to a, a3 and a3+3a2c+3ac2+c are the cubes of the supposed roots. Now, if we double the first cube (a3), and add it to the second, we shall have 3a3+3a2c+3ac2+c3 If we double the second cube and add it to the first, we shall have 3a3+6a2c+6ac2+2c3 As c is a very small fraction compared to a, the terms containing c2 and c3 are very small in relation to the others; and the relation of these two sums will not be materially changed by rejecting those terms containing c2 and c3, and the sums will then be 3a3+3a2c And The ratio of these terms is the same as the ratio of a+c to a+2c. But the ratio of the roots a to a+c, is 1+. Observing again, that c is supposed to be very small in relation to a, the fractional parts of the ratios and are с a+c с a both small, and very near in value to each other. Hence, we have found an operation on two cubes which are near each other in magnitude, that will give a proportion very near in proportion to their roots; and by knowing the root of one of the cubes, by this ratio we can find the other. For example, let it be required to find the cube root of 28, true to 4 or 5 places of decimals. As we wish to find the cube root of 28, we may assume that 28 is a cube. 27 is a cube near in value to 28, and the root of 27 we know to be 3. Hence, a, in our investigation, corresponds to 3 in this example, and c is unknown; but the cube of a+c is 28, and a3 is 27. Or, (a+c)=24=3.03658+, which is the cube root of 28, true to 5 places of decimals. By the laws of proportion, which we hope more fully to investigate in a subsequent part of his work, the above proportion, 8283a:a+c, may take this change, 82: 1::a:c Hence, c; c being a correction to the known root, which, being applied, will give the unknown or sought root. For what precedes, we may draw the following rule for finding approximate cube roots: RULE. Take the nearest rational cube to the given number, or assume a root and cube it. Double this cube, and add the number to it; also double the number and add the assumed cube to it. Then, by proportion, as the first sum is to the second, so is the known root to the required root. EXAMPLES. 1. What is the approximate cube root of 122? |