248. These operations may be explained in the following manner; I. Since the cube root of 1000 is 10, of 1000000 is 100, &c.; it follows, that the cube root of a number less than 1000 will consist of one figure; of a number between 1000 and 1000000 of two figures, &c. &c.; if, therefore, the given number be divided into periods, each consisting of three figures, by placing a dot over every third figure beginning with the units, the number of those dots will show the number of figures of which the cube root consists; and for the reason assigned in the preceding Article, (respecting the first figure of the square root), the first figure of the root will be the cube root of the greatest cube number contained in the first period. II. Having pointed the number, we find that its cube root consists of three figures. The first figure is the cube root of the greatest cube number contained in 13; this being 2, the value of this figure is 200, or a=200, consequently a= 8000000; subtract this number from 13997521, and the remainder is 5997521. Find the value of 3a2, and divide this latter number by it, and it gives 40 for the value of b, the second number of the root; put this in the quotient, and then calculate the value of 3ab+3ab2+b3 and subtract it, and there remains 173521. Find now the value of 3×(a+b)3, and divide 173521 by it, and it gives 1 for the value of c, the third member of the root; put this in the quotient, and then calculate the amount of 3(a+b)c+3(a+b)c2+c3, which subtract, and nothing remains. III. In reviewing the first of these two operations, it is evident that six ciphers might have been rejected in the value of a3, and three in the value of 3a2b+3ab2+b3, without af fecting the substance of the operation; having therefore simplified the process as in the second operation, we are furnished with the following rule, for extracting the cube root of numbers. RULE. 249. Point off every third figure, beginning with the units; find the greatest cube number contained in the first period, and place the cube root of it in the quotient. Subtract its cube from the first period, and bring down the next three figures; divide the number thus brought down by 300 times the square of the first figure of the root, and it will give the second figure; add 300 times the square of the first figure, 30 times the product of the first and second figures, and the square of the second figure together, for a divisor; then multiply this divisor by the second figure, and subtract the result from the dividend, and then bring down the next period, and so proceed till all the periods are brought down. The rules for extracting the higher powers of numbers, and of compound algebraic quantities, are very tedious, and of no great practical utility. Examples for practice in the Square and Cube Roots of Ex. 1. Required the square root of 106929. Ex. 2. Required the cube root of 48228544. Ex. 3. Required the square root of 152399025. Ex. 4. Required the square root of 5499025. Ex. 5. Required the cube root of 389017. Ans. 12345. Ans. 2345. Ans. 73. Ans. 103. 198 CHAPTER VII. ON IRRATIONAL AND IMAGINARY QUANTITIES. § I. THEORY OF IRRATIONAL QUANTITIES. P 250. It has been demonstrated (Art. 231), that the mth root of a", the exponent p of the power being exactly divisible by the index m of the root, is am. Now in case that the exponent p of the power is not divisible by the index m of the root to be extracted, it appears very natural to employ still the same method of notation, since that it only indicates a division which cannot be performed: then the root cannot be obtained, but its approximate value may be determined to any degree of exactness. These fractional exponents will therefore denote imperfect powers with respect to the roots to be extracted; and quantities, having fractional exponents, are called irrational quantities, or surds. It may be observed that the numerator of the exponent shows the power to which the quantity is to be raised, and the m denominator its root. Thus, an is the nth root of the mth power of a, and is usually read a in the power (). 251. In order to indicate any root to be extracted, the radical sign is used, which is nothing else but the initial of the word root, deformed, it is placed over the power, and in the opening of which the index m of the root to be extracted is written. P We have therefore /a" am. = For the square root, the sign is used without the index 2; thus, the square root of ar is written ar, as has been already observed, (Art. 18). Quantities having the radical sign prefixed to them, are called radical quantities: thus, a, b, c, V/m, &c. are radical quantities; they are, also, commonly called Surds. 252. From the two preceding articles, and the rules given in' the second section of the foregoing Chapter, we shall, in general, have, P 오 go W/(ar.bq.cr)=W/aP >~/ba ×/cr=am xbmxcm ; /ar.b¶__m/(ar.bq).__"/br>W/bq m = c'd /c'd P 253. Two or more radical quantities, having the same index, are said to be of the same denomination, or kind; and they are of different denominations, when they have different indices. In this last case, we can sometimes bring them to the same denomination; this is what takes place with respect to the 2 two following, ab and 1/a*b*=aa×b*=a*.ba = √a^√/b2 = ab. In like manner, the radical quantities 3/2ab and 3/16ab, may be reduced to other equivalent ones, having the same radical quantity; thus, 3/2a*b=3/a°×3/2b=a23/2b, and 2/16ab=3/8a2b=3/8. /a3. 3/2b=2a3/2b; where the radical factor 3/26 is common to both. 254. The addition and subtraction of radical quantities can in general be only indicated : Thus, a added to, or subtracted from ✅b, is written b a2, and no farther reduction can be made, unless we assign numeral values to a and b. But the sum of a2b, a2b, and √4ab is ab+ab+2a√/b=4a/b; 32/ab/ab=2 ab; and (b+ab)Noa. ab2+/ab1=bia+aba2=ba+ab1/a= 255. Hence we may conclude, that the addition and subtraction of radical quantities, having the same radical part, are performed like rational quantities. Radical quantities are said to have the same radical part, when like quantities are placed under the same radical sign; in which case radical quantities are similar or like. It is sometimes necessary to simplify the radical quantities, (Art. 252), in order to discover this similitude, and it is independent of the coefficients. Thus, for example, the radical quantities 363/2a5b2, 8a3/ 2ab5, and -7ab3/2ab2, become, by reduction, 3ab3/2a2b2, Sab/2ab2, and -7ab3/2a2b2; which are similar quantities, and their sum is =4ab3/2a2b2. 256. We have demonstrated, (Art. 252), this formula,*/ arbicr=m/arm/bq/cr; from which the rule for the mul tiplication of radical quantities, under the same radical sign, may be easily deduced. 257. Let us pass to radical quantities with different indices, and suppose that we had to find, for instance, the product of P m/ap by m/bq, or that of am by bm: we can bring this case to the preceding, by reducing to the same denominator, (Art. 152), the fractions and ; and we shall have/arm/b9 p m PL pm' qm 9 m =ambm' = a mm2 ×bmm' = mm'/apm' ×mm' 'Xbmm' = mm/apm' mm/bqm = mm2/apm'bqm. 258. The rule for dividing two radical quantities of the same kind, may be read in this formula (Art. 233.) /ap m bq and it only remains to extend it to two radical quantities of different denominations. Let therefore /ar be divided by m/ba: by passing from radical signs to fractional exponents, we have m P pm/ Var am amm We may likewise suppose, under the radical signs, any number of factors whatever, and it shall be easy to assign the quotient, (Art. 252). Let now a bin the formula m/arm/bq=m/ap. bq; it becomes, by passing from radical signs to fractional exponents, amXam=m/aP+¶=a m=qm*m Therefore the rule demonstrated (Art. 71), with regard to whole positive exponents, extends to fractional exponents. n/ar 259. In the same hypotheses b=a, the quotient be another extension of the rule given (Art. 86), to fractional positive exponents. |