Thus, (3)=2x2x2=2X2X2 3 3 3×3×3 3° 27 23 8 ; 3x 64 27 8 27 240. 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. Thus, 3 5×72 5×52 5×72 = 3/5x73 = 3/5x72 = √5x72; &c. As a general 7×72 5×72 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, 8 is not 2; the cube of 2 being (2 × 2x2 =) *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' 80,000,000 800,000,000, &c. Rejecting the fractions ( 100,000,000' 1,000,000,000' 80 8,000 80,000 8,000,000 80,000,000 8 100' 10,000' 100,000' 10,000,000' 100,000,000' &c.) whose denominators are not cube numbers, we have V-8: $800,000 or 100 800,000,000 &c.; 7.8 9 or 92 or 929 ΙΟ 100' 1,000* &c. ; V·8='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 V6543 Converting into a decimal, we have V6543 8 17 8 ing is carried to six places: 6543 6543′470588 = 17 3/6543470588 1,000,000 closer approximation is obtained when the decimal representing 8 is carried to nine places: 65436543 470588235= 17 3/6543470588235 1,000,000,000 18.712. 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 57 by less than we multiply 57 by 123, and divide 12 into the *92456 lies between 12 46 12 and 47; and as the last two fractions 12 differ by exactly —, each differs from 57 by less than 12 1 12 Again, to find an approximation which shall differ from 4567 by less than we multiply 4567 by 1,0co', and divide 1.000 into the cube root of the product: 4567 1,000' = 3/4567 3 / 4567 x 1.005 - 3 / 4567,000,000,000 - $4567,000,000,000, 1,0002 = 3/4567. 10001 1,000 not differ from either 16.593 or 16.594 by so much as or '001. I 1,000' I 1,000' Instead of saying that an approximation is to differ from a required root by less than we usually say that the root is to be "true" to three decimal places; and, in practice, 4567 would be written under the form 4567 000,000,000-the resulting approximation differing from the truth by less than what the digit I would represent in the last decimal place of the approximation. 243. Knowing how to extract square roots and cube roots, we can (in a round-about way) extract such other roots as the 4th, 6th, 8th, 9th, 12th, &c.-in fact, any root indicated by a number which contains no prime factor different from 2 and 3. For instance: The square root of of a number is the 4th root of the number Thus, the square root of a1 is x2, whose square root, x, is the 4 root of a; the square root of x is z3, whose cube root, x, is the 6th root of 26; the 4th root of 28 is x2, whose square root, r, is the 8th root of r; the cube root of x9 is x3, whose cube root, x, is the 9th root of r9; the 4th root of x12 is x3, whose cube root, x, is the 12th root of 212; &c. 244. Fractional Indices.-The employment of fractional indices enables us to write roots under the form of powers-a very convenient form in many cases. Thus, instead of Because (§ 220) the square of a3 is a; the cube of b3 is b; the fourth power of c is c; the square of x is r3; the cube of y3 is y2; &c.: (a)2=a1×a1=a1+1_a1= a; (b13 = b + xbx b$ = b $ + b + + = b2 = b; (c)'=c*xc*xc*xc*= c + + + + + + 3 ({ x})2=x3×xl=x1+1=x3; (y3)3=y3×y*×y3=y#+8+8 = c12 = c; =y2; &c. The expressions a, b, and ct would be read "a in the power," "b in the power," and "e in the power 1," respectively. Such an expression as x or y-the numerator of the index not being unity-could be read in either of two ways: ris "the square root of the cube of x," or "the cube of the square root of x;" and y3 is "the cube root of the square of y,” of the cube root of y." square It has already been shown that is the square root of 23, and y3 the cube root of y; and it can be shown, quite as easily, that is the cube of x, and y3 the square of y: (x)=x Xxxx+= x + + + + + _xi; (y*)2=y*\y*=y* „j+š—y3. .or "the PROGRESSION. 3 245. By a PROGRESSION is meant a series of three or more numbers which successively increase, or successively decrease, at some uniform rate. 246. The numbers forming a progression are called its TERMS, of which the first and the last are known as the extremes, and the intermediate terms as the means. 247. A progression is said to be ARITHMETICAL (or equidifferent) when, of every three consecutive terms, the difference between the first and the second is equal to the difference between the second and the third; GEOMETRICAL (or equirational), when, of every three consecutive terms, the ratio which the first bears to the second is equal to the ratio which the second bears to the third; and HARMONICAL, when, of every three consecutive terms, the first bears to the third the same ratio which the difference between the first and the second bears to the difference between the second and the third. 248. A progression is called an ascending or a descending one-according as the terms increase or decrease from left to right. ARITHMETICAL PROGRESSION. 249. Being given the first term and the common difference, we can form an arithmetical progression by continually adding the common difference to, or continually subtracting it from, the first term-according as the progression is "ascending" "descending." or EXAMPLE I.-Set down the ascending arithmetical progression which has I for its first term, and 2 for common difference. Adding 2 to 1, we obtain the second term, 3; adding 2 to 3, we obtain the third term, 5; adding 2 to 5, we obtain the fourth term, 7; and so on. The progression, therefore, isI 3 5 7 9 II 13 15 17 19 21 23 &c. EXAMPLE II-Set down the descending arithmetical progression which has 35 for its first term, and 3 for common difference. Subtracting 3 from 35, we obtain the second term, 32; subtracting 3 from 32, we obtain the third term, 29; subtracting 3 from 29, we obtain the fourth term, 26; and so on. The progression, therefore, is 35 32 29 26 23 20 17 14 II 8 5 2 From an examination of the preceding examples it will be |