annexing two ciphers; convert the dividend into a new dividend, by bringing down the next period ; and convert this new dividend into a new trial-dividend by conceiving the last two figures cut off. (234). NOTE 2.-When the “ discovered” figures of a cube root are greater in number, by 2, than those remaining to be discovered, the latter can be found by division only, as follows: For dividend, bring down the outstanding periods, and cut off twice as many figures as there are root-figures to be determined; for divisor, treble the square of the known part of the root; the resulting quotient —the remainder being disregarded—will be the rest of the root. Thus, in extracting V 12906416731998560897069, we need not work, in the ordinary way, for more than the first five rootfigures (23456) — knowing that the total number is eight +1 After the finding of the fifth root-figure (6), we have merely to bring down the outstanding periods--all but the last six figures; to divide 1302329182560, the number so obtained, by (234562x3=)1650551808; and, disregarding the remainder, to write 789--the resulting quotient—as the rest of the root : 1290641673i998560897069 | 23456 8 63 4906 3 4167 189 739416 4 645904 1389 2776 7025 93512731 3 1587 5 82309625 4167 161476 35125 11203106998 4 164268 9900777816 645904 16461925 1302329182 5 70356 6 82309625 | 694 12 422136 16497075 1650129636 234562 x 3=550183936 x 3=1650551808; 1302329182560: 1650551808 = 789; re quired root=23456789. 9900777816 . 22 10 The reason is this: Putting x for the last three figures of the root, we have 12906416731998 560897069=23456000+ X; 12906416731998560897069= (23456000+x)=234560003 +3 x 234560002 x 2 +3 x 23456000 x x2 +2%. Subtracting 234560003, therefore, from 12906416731998560897069, we ob. tain 1302329182560897069=3 x 234560002 xx+3x23456000 = =x + + 3 X 23456000? 3 X 23456000 23 x2 X3 As the num3 x 234560002 23456000 3X234560002 bers represented by x occupies only three places, that represented by x2 occupies not more than six places ($ 230), and is x2 therefore less than 1,000,000; so that is less than 23456000 1,000,000 But : because 23,456,000 23,456,000 10,000,000 23,456,000 occupies eight places, and the smallest (whole) number which cccupies eight places is 10,000,000. Consequently, is less than 1,000,000 or than I. Again: 23456000 10,000,000' the number represented by x occupies not more than nine places (§ 237), and is therefore less than 1,000,000,000; so that is less than But 1,000,000,000 3 X 234560002 3x 23456000 3X234560002 is less than 1,000,000,000 : because the square of 100,000,000,000,000 23456000 occupies at least fifteen places ($ 230), and the smallest (whole) number which occupies fifteen places is. 23 100,000,000,000,000. Consequently, is less than 3X23456000 1,000,000,000 It thus appears that 100,000,000,000,000' 100,000 x2 r3 and therefore 23456000 3 X 234560002 100,000 than unity. So that the division of 1302329182560897069 by 3x 234560002 —or of 1302329182560897069 by 1650551808000ooo, or of 1302329182560 by 1650551808-gives x, the remainder being disregarded. 239. As the cube of a fraction is obtained when the cube of the numerator is divided by the cube of the denominator-so, the cube root of a fraction is obtained when the cube root of the numerator is divided by the cube root of the denominator. I or than I I IO * 27 = 3; &c. = a 3 *** = *4*5?; 3 7644 . 240. The cube root of a fraction whose terms are not cube numbers, or the cube roots of whose terms cannot be found by inspection, is obtained with comparative facility when we multiply the numerator by the square of the denominator, and divide the denominator into the cube root of the product. 4x52 4x52 Thus, 58 5 x 72 5x72 35x7 &c. As a general 7x72 7 rule, however, the cube root of a fraction is most easily extracted after the fraction has been converted into a decimal ($ 241). 241. To extract the Cube Root of a number in which a Decimal occurs: Treat the number as an integer-having (if necessary) first annexed a cipher or two, in order to make the number of decimal places a multiple of 3; and remove the decimal point in the resulting root a place to the left for every three decimal places in the given number. As an illustration, let it be required to extract 8. Although 8 is 2, V-8 is not '2; the cube of "2 being ( 2 X 2 X 2 =) •008. We can write :8 under any of the following forms : 8 80 800 8,000 80,000 800,000 8,000,000 10'100' 1,000' 10,000' 100,000' 1,000,000' 10,000,000 8 100,000,000' 1,000,000,000' 80,000,000 800,000,000 , &c. Rejecting the fractions (o &c.) whose , 3 ♡ V 3 800 1,000 3 3 800 or or IO or or or 100 &c.; 3 V IO 10 V800.000 9 92 929 &c.; 1,000 100' 1,000' V8=9, or '92, or •929, &c. In practice, -8 would be written :800, or :800,000, or .800,000,000, &c.—according to the degree of accuracy considered necessary : in the extraction of its cube root, the number would then be treated as an integer, and the decimal point placed before the first root-figure (9). As a second illustration, let it be required to extract 8 *65431. Converting into a decimal, we have 0654317 17 6543470 6543470 187 06543-47 o= 18.7. 1,000 A closer approximation is obtained when the decimal represent 8 ing is carried to six places: 6543175 76543-470588= 17 3/6543470588 76543470588_1871 18.71. A still 1,000,000 closer approximation is obtained when the decimal representing is carried to nine places : 165431 = V6543'470588235= 17 3 6543470588235 56543470588235_18712 18.712. 1,000,000,000 1,000 1,000 In practice, the given number would be written 6543.470, or 6543-470588, or 6543-470588235, &c.—the decimal being made to occupy three places for every figure required in the decimal part of the root: the number would then be treated as an integer,and the decimal point placed before 7—the first root-figure obtained after the bringing down of the first period (470) from the decimal part of the given number. 242. To find an Approximation which shall differ from the Cube Root of a number by less than a given Fractional Unit: Multiply the number by the cube of the denominator of the fractional unit, and divide the denominator into the cube root of the product. Thus, to find an approximation which shall differ from 357 by less than we multiply 57 by 12', and divide 12 into the cube root of the product: V57= 57 x 1728 123 12 98496 98496. As 98496 lies between 46 and 47, 123 I 12 3 57 x 123 3 12 *98496 lies between ܪ I2 I 12 I 3 3/45675 i and I I or 'OOI. I 46 and as the last two fractions 12 12 I differ by exactly in each differs from 57 by less than 12' Again, to find an approximation which shall differ from 14567 by less than we multiply 4567 by 1,000, and 1,000' divide 1,000 into the cube root of the product: 34567 = 3/4567 X 1.000* 4567,000,000,000 4567,000,000,000 1,0003 10003 1,000 As 4567,000,000,000 lies between 16593 and 16594, 4567,000,000,000 must lie between 16593 and 16594 1,000 1,000 1,000 as the last two fractions differ by exactly 1,000' 84567 does not differ from either 16.593 or 16.594 by so much as 1,000 Instead of saying that an approximation is to differ from a required root by less than we usually say that the root 1,000' is to be “true" to three decimal places; and, in practice, 4567 would be written under the form 4567*000,000,000—the resulting approximation differing from the truth by less than what the digit I would represent in the last decimal place of the approximation. 243. Knowing how to extract square roots and cube roots, we can (in a round-about way) extract such other roots as the 4th, 6th, 8th, 9th, 12th, &c.—in fact, any root indicated by a number which contains no prime factor different from 2 and 3. For instance: The square root of the square root The cube root of 6th the square root The square root of 8th the 4th root The cube root of the cube root The cube root of the 4th root &c. &c. Thus, the square root of x4 is xa, whose square root, x, is the 4 root of x4; the square root of ze is z', whose cube root, x, }of a number is the 4" root of the number : |