in 4 once, 1 is the third figure of the root; place 1 therefore both in the divisor and quotient; multiply and subtract as before, and nothing remains. 308. The method of extracting the cube root of numbers may be understood by comparing the process for extracting the cube root of (a+b+c)3, (Art. 300), with the following operations, in which is deduced the cube root of the number 13997521. 13997521(200+40+1 a3 (200)3=8000000 = 1st remainder 5997521 3u2=3X(200) divisor, ... 3a2b=3(200) x 40=4800000 = = 3(a+b)'c=(200+40) x1=172800 3(a+b)c2=3(200+40)×1= 720 Omitting the superfluous ciphers, and bringing down three figures at a time, the operation will stand thus; 309. These operations may be explained in the following manner; Snce 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 a3=8000000; subtract this number from 13997521, and the remainder is 6997521. 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 3a2b+3ab2+b3 and subtract it, and there remains 173521. Find now the value of 3 × (a+b)2, 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)2c+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 affecting 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. 310. 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 Ex. 1. Required the square root of 106929. 9 62 169 647 4529 4529 Ex. 2. Required the cube root of 48228544. Ex. 3. Required the square root of 152399025. Ans. 12345. Ex. 4. Required the square root of 5499025. Ans. 2345. Ex. 5. Required the cube root of 389017. Ans. 73. Ex. 6. Required the cube root of 1092727. Ans. 103. CHAPTER VII. ON IRRATIONAL AND IMAGINARY § I. THEORY OF IRRATIONAL QUANTITIES. 311. It has been demonstrated (Art. 292), that the mth root of ao, the exponent p of the power being P 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 frac tional 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 denominator its root. Thus, an is the nth root of the mth power of a, and is usually read a in the power 312. 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. |