5. To find the third power of 2 a2 — 4 a b + 3 b2. SECTION XVII. —ROOTS OF COMPOUND QUANTITIES. 136. We pass next to the extraction of the roots of compound quantities, beginning with the third or cube root of numbers. In the following table, we have the nine first numbers, with their third powers or cubes written under them respectively. 1, 2, 3, 4, 5, 6, 7, 8, 9 1, 8, 27, 64, 125, 216, 343, 512, 729 By inspection of this table, it will be perceived, that among numbers consisting of two or three figures, there are nine only, which are perfect third powers, the others have each for a root an entire number plus a fraction. If the proposed number consists of not more than three figures, its third root, or that of the greatest third power contained in it, may be found immediately by the above table. Let it be proposed to extract the third root of a number, consisting of more than three figures, 103823, for example. The proposed being comprised between 1000, the third power of 10, and 1000000, the third power of 100, its root will consist of two places, units and tens. To return therefore from the proposed to its root, let us observe the manner, in which the units and tens of a number are employed in forming the third power of this number. For this purpose designating the tens by a and the units by b, we have (a+b)3 = a3 + 3 a2 b + 3 a b2 + b3 From this we learn, that the third power of a number consisting of units and tens is composed of the third power of the tens, the triple product of the square of the tens by the units, the triple product of the lens by the square of the units, and the third power of the units. If then we can determine in the proposed the third power of the tens, the tens of the root will be found by extracting the third root of this part. The third power of the tens, it is evident, can have no significant figure below the fourth place, the three figures on the right will therefore form no part of the third power of the tens, and may on this account be separated from the rest by a comma. The third power of the tens will then be contained in 103, the part at the left of the comma. The greatest third power contained in 103 is 64, the root of which is 4; 4 is therefore the significant figure in the tens of the root sought. Indeed, the proposed is evidently comprised between 64000, the third power of 40 or 4 tens, and 125000 the third power of 50 or 5 tens. The root sought is therefore composed of 4 tens and a certain number of units less than ten. The tens of the root being thus obtained, we subtract the third power 64 from 103, the part of the proposed at the left of the comma, and to the remainder bring down the figures at the right. The result of this operation, 39823, must contain, from what has been said, the triple product of the square of the tens by the units together with the two remaining parts in the third power of the root sought. The square of the tens, it is evident, will contain no significant figure less than hundreds, on this account we separate 23, the two figures on the right of the remainder 39823, from the rest by a comma; 398, the figures on the left of the comma, will then contain the triple product of the square of the tens of the root sought by the units and something more, in consequence of the hundreds arising from the two remaining parts of the third power of the root sought. Dividing therefore 398 by 48, the triple product of the square of the tens, already found, the quotient 8 will be the unit figure sought, or, from what has been said, it may be too large by 1 or 2. To determine whether 8 be the right unit figure we raise 48 to the third power. This gives 110592, a number greater than the proposed; 8 is therefore too large for the unit figure. We next try 7; 47 raised to the third power gives 103823. The proposed is therefore a perfect third power, the root of which is 47. The operation may be exhibited as follows. 103,823 | 47 64 398,23 | 48 103,823 Any number however large may be considered as composed of units and tens; the process for finding the root may therefore be reduced to that of the preceding example. Let it be proposed, for example, to find the third root of 43725678. Considering the root of this number as composed of units and tens, 678 the three right hand figures, it is evident will form no part of the third power of the tens. On this account we separate them from the rest by a comma. The third power of the tens being contained then in the part at the left of the comma, we obtain the tens of the root sought by extracting the third root of this part. Considering therefore, for the moment, the part of the proposed 43725 as a separate number, its third root, it is evident, may be found as in the preceding example. Performing the operations, we have 35 for the root and a remainder of 850. There will therefore be 35 tens in the root of the proposed, and in order to find the units, we bring down the three right hand figures 678 by the side of 850, which gives 850678. Separating next the two right hand figures of this last from the rest by a comma, and dividing the part on the left by the triple square of the tens already found, we obtain 2 for the unit figure of the root sought. To determine whether this is the right figure, we raise 352 to the third power, which gives 43614208, a result less than the proposed. .352 is therefore the root of the proposed to within less than a unit. The operation may be exhibited as follows 43,725,678 | 352 27 167,25 | 27 1st Divisor 8506,78 3675, 2d Divisor 43614208 The same process, it is easy to see, may be extended to any number however large. The rule therefore for the extraction of the third root will be readily inferred. If it happens, that the divisor is not contained in the dividend prepared as above, a zero must be placed in the root, and the next figure brought down to form the dividend. EXAMPLES. 1. To find the third root of 91632508641. 2. To find the third root of 32977340218432. 3. To find the third root of 217125148004864. 137. If the proposed be a fraction its third root is found by extracting the third root of the numerator and denominator. If the denominator is not a perfect third power it may be made so, by multiplying both terms by the square of the denominator; thus if the proposed bewe multiply both terms by 49; the fraction then becomes 5 3 147 343' nearest, accurate to within less than the root of which is 138. We have seen, art. 97, that the square root of an entire number, which is not a perfect square, cannot be exactly assigned. The same is true with respect to the roots of all entire numbers, which are not perfect powers of a degree denoted by the index of the root. The third root of a number, which is not a perfect third power may be approximated by converting the number into a fraction, the denominator of which is a perfect third power. Thus let it be required to find the approximate third root of 15. This 15 × 123 25920 number may be put under the form the 1:23 accurate to within less than 1728 If a greater degree of accuracy were required, we should convert the proposed into a fraction, the denominator of which is the third power of some number greater than 12. In such cases it is most convenient to convert the proposed number into a fraction, the denominator of which shall be the third power of 10, 100, 1000, &c. Thus if it be required to find the third root of 25 to within .001, we convert the proposed into a decimal, the denominator of which is the third power of 1000, viz. 25.000000000, the third root of which is 2.920 to within .001; we have then 3/252.920 accurate to within less than .001. To approximate therefore the third root of an entire number by means of decimals, we annex to the proposed three times as many zeros as there are decimal places required in the root, we then extract the root of the number thus prepared to within a unit, and point off for decimals, as many places as there are decimal figures required in the root. 139. If the proposed number contain decimals, beginning at the place of units, we separate the number both to the right and left into periods of three figures each, annexing zeros if necessary to complete the right hand period in the decimal part. We then extract the root, and point off for decimals in the root as many places as there are periods in the decimal part of the power. If the proposed be a vulgar fraction, the most simple method of finding the third root is to convert the proposed into a decimal, the number of places in which shall be equal to three times the number of decimal figures required in the root. The question is thus reduced to extract the third root of a decimal fraction. EXAMPLES. 1. To find the approximate third root of 79. 140. By processes altogether similar to that, which we have employed in the extraction of the third root of numbers, we may extract the root of any degree whatever. The method of extracting the root of any degree whatever, in the case of algebraic quantities, is also founded upon the same principles. |