membered 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. Afterwards 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. (Art. 78.) The methods of direct extraction of the cul e 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+c2 are the cubes of the supposed roots. Now if we double the first cube (a3), and add it to the second, we shall have За3+3a2c+3ac2+c2. If we double the second cube and add it to the first, we shall have За3+6a2c+6ас2+2c3. As c is a very small fraction compared to a, the terms containing ca 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 ca and c3, and the sums will then be And 3a3+3a2c The ratio of these terms is the same as the ratio of atc to a+2c. Or the ratio is But the ratio of the roots a to a+c, is 1+ C a Observing again that c is supposed to be very small in C relation to a, the fractional parts of the ratios and atc C a are 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 cis unknown; but the cube of ac is 28, and a3 is 27. Sums 82 : 83 :: 3 : atc very nearly. Or 28, true to 5 places of decimals. (a+c)==3.03658+, which is the cube root of By the laws of proportion, which we hope more fully to investigate in a subsequent part of this work, the above proportion, 82:83::a:a+c, may take this change: known root, which, being applied, will give the unknown or sought root. From what precedes, we may draw the following rule for finding approximate cube roots: RULE. Take the nearest rational cube to the given number, and call it an assumed cube; or, assume a root to the given number and cube it. Double the assumed cube and add the given number to it; also, double the given 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. Or take the difference of these sums, then say, as double of the assumed cube, added to the number, is to this difference, so is the assumed root to a correction. This correction, added to or subtracted from the assumed root, as the case may require, will give the cube root very nearly. By repeating the operation with the root last found as an assumed root, we may obtain results to any degree of exactness; one operation, however, is generally sufficient. EXAMPLES. 1. What is the approximate cube root of 120% 1 Ans. 4.93242+ 2. What is the approximate cube root of 85? Ans. 2.0408+ Ans. 3.97905+ 3. What is the approximate cube root of 6 3? 4. What is the approximate cube root of 515? Ans. 8.01559+ 5. What is the approximate cube root of 16? The cube root of 8 is 2, and of 27 is 3; therefore the cube root of 16 is between 2 and 3. Suppose it 2.5. The cube of this root is 15.625, which shows that the cube root of 16 is a little more than 2.5, and by the rule Approximate root 2.51984 We give the above as an example to be followed in most cases where the root is about midway between two integer numbers. This rule may be used with advantage to extract the root of perfect cubes, when the powers are very large. EXAMPLE. The number 22.069.810.125 is a cube; required its root. Dividing this cube into periods, we find that the root must contain 4 figures, and the superior period is 22, and the cube root of 22 is near 3, and of course the whole root near 3000; but it is not 3000. Suppose it 2800, and cube this number. The cube is 21952000000, which being less than the given number, shows that our assumed root is not large enough. To apply the rule, it will be sufficient to take six superior figures of the given and assumed cubes. Then by the rule 11* 659738)3298400(5 The result of the last proportion is not exactly 5, as will be seen by inspecting the work; the slight imperfection arises from the rule being approximate, not perfect. When we have cubes, however, we can always decide the unit figure by inspection, and, in the present example, the unit figure in the cube being 5, the unit figure in the root must be 5, as no other figure when cubed will give 5 in the place of units. [For several other abbreviations and expedients in extracting cube root in numerals, see Robinson's Arithmetic.] (Art. 79.) To obtain the 4th root, we may extract the square root of the square root. To obtain the 6th root, we may take the square root first, and then the cube root of that quantity. To extract odd roots of high powers in numeral quantities is very tedious and of no practical utility; we therefore give no examples. (Art. 80.) Roots of quantities may be merely expressed by radical signs. For example, the cube root of 16 may be expressed thus: 16, or 165. If a cube factor is under the sign, that factor may be taken out by putting its root as a multiplier without the sign. In this example 16 has the cube factor 8, and 16=8x2=2√2. That is, twice the cube root of 2 is equal to the cube root of 16. Hence if the root of 2 is known, the root of 16 is equally known. The cube root of 40 is '√40='√8×5=2*√5. 3 3 In the same manner we may express the square root of any number. Thus, the square root of 18 is √18= √9×2=3√2. The square root of 24 is 2√6. Observe that we pick out the square or cube factors, as |