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. 106929(327 9 62 169 647 4529 4529 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. Ang. 103. CHAPTER VII. ON IRRATIONAL AND IMAGINARY QUANTITIES. § I. THEORY OF IRRATIONAL QUANTITIES. 311. It has been demonstrated (Art. 292), that the mth root of ap, the exponent p of the power being exactly divisible by P the index m of the root, is am. Now in case that the expo-, nent 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, a " 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 ra dical 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 "/a2=a". For the square root, the signis 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, v/c2, vxm, &c. are radical quantities; they are, also, commonly called Surds. 313. From the two preceding articles, and the rules given in the second section of the foregoing Chapter, we shall, in general, have, */(ap.bq.c2)="/ap×m/b9×"/c=aTM×bm×cm ; 3 cxbm Therefore, ab=3/a3×3/b=a × 3/b=a3/b ; = ex3/xz ex = = 314. 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 4 3 2 two following, a3b2 and ÷/ab1—aa ×b*=a3· .b3 = √ a3b2= ✔a3b3. In like manner, the radical quantities 3/2ab and 3/ 16a3, may be reduced to other equivalent ones, having the same radical quantity; thus, /2ab=3′a®×3/2b=a23/2b, and 3/16a3b=3/8a3.2b=3/8.3/ a3. 3/ 2b=2u3/2b; where the radical factor 3/26 is common to both. 315. The addition ard subtraction of radical quantities can in general be only indicated: Thus, /a2 added to, or subtracted from √b, is written b +a2, and no farther reduction car. be made, unless we assign numeral values to a and b. But the sum of √✅a2b, √a2b, and ab+a√b+2a√b=4a/b; 32/ab/ab=2 ab2+/ab1 = ba+aba2 = b√ataba= √4ab is 316. 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. 313), in order to discover this similitude, and it is independent of the coefficients. Thus, for example, the radical quantities 363/2a5b2, 8a 2a2b5, and 7ab2/2a2b2, become, by reduction, 3ab2a2b2, 8ab2/2ab2, and -7ab3/2a2b2; which are similar quantities, and their sum is =4ab3/2a2b2. 317. We have demonstrated, (Art. 313), this formula, m/ arbicTM="/apX/b9X/c"; from which the rule for the multiplication of radical quantities, under the same radical sign, may be easily deduced. 318. Let us pass to radical quantities with different indices, and suppose that we had to find, for instance, the product of P /a" by m/b9, or that of am by b: we can bring this case to the preceding, by reducing to the same denominator, (Art. 152), the fractions, and ; and we shall have "/ap×TM/b9 P 9 = amma pm mm' Xbmm2 m 1m2/apm' × mm2/bqm = mm/qpm'fqm. 319. The rule for dividing two radical quantities of the same kind, may be read in this formula (Art. 294.) m m ap and it only remains to extend it to two radical quantities of different denominations. Let therefore mar be divided by m/ba: by passing from radical signs to fractional exponents, we have We may likewise suppose, under the radical signs, any number of factors whatever, and it shall be easy to assign the quotient, (Art. 313). Let now a=b in the formula it becomes, by passing from radical signs to fractional exponents, P 9 ገዜ Therefore the rule demonstrated (Art. 71), with regard to whole positive exponents, extends to fractional exponents. 320. In the same hypothesis ba, the quotient be am another extension of the rule given (Art. 86), to fractional positive exponents. 321. We may, in the preceding formula, suppose p=o; and demonstrated, (Art. 86), in the case of whole exponents, and which still takes place when the exponents are fractional. 322. If we now admit the two equalities, and if we multiply them member by member, we shall have the equal products, It appears therefore evident, that exponentials, with fractional negative exponents, follow the same rule in their multiplication, as those with whole positive exponents. 323. The division of am, by am, gives for the quotient Now the exponent of the quotient, namely, is the exponent of the dividend, minus that of the divisor, which is still a generality of the rule (Art. 86), relative to the division of exponentials. 324. The rules that have been demonstrated in the preceding articles may be extended to radical quantities having irrational exponents: For instance a√2, 3, &c. since that the roots of 2 and 3 might be obtained with a sufficient degree of approximation, and such that the error may be neglected; so that these exponents shall be terminated decimal fractions, which can be always replaced by ordinary fractions. 325. The formation of the powers of radical quantities, is nothing else but the multiplication of a number of radical quantities of the same denomination, marked by the degree of the power; so that it is sufficient to raise the quantity under the radical sign to the proposed power, and afterwards |