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 tigure, 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 quaotities, are very tedious, and of no great practical utility. Examples for practice in the Square and Cube Roots of Numbers. 106929(327 Ex. 2. Required the cube root of 48228544. 48228544(364 3276)21228 19656 Divide by 300 X 39=2700 30 X3X6= 549 6 X 6= 36 393136) 1572544 157 2544 1st Divisore 3876 Divide by, (36)2 X 300=388800 30 X 36 X4= 4320 4 X4= 16 2d Divisor 393136 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. Aas. 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 a"Now in case that the expo. dent р 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. lo order to indicate any root to be extracted, the ra: dical sigo 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. We have therefore/ap=a". For the square root, the sign is used without the index 2; thus, the square root of ar is written vap, as has been already observed, (Art. 18). Quantities having the radical sign V prefixed to them, are called radical quantities : thus, Va, vb, vc, x, &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, 9 ex 4 2 P q m (ap.68.c*)="/apxm, 69xm/c=am X bm Xcm; "Var.b?_V(a”.6?) _Vbox/b9_amx bm c*ds cdo mcXmds I стX, т Therefore, Vab=»a'x/b=ax yb=a7b; and 2%*c* _Vabc-Va® Xybx e'x*z Ve'x'z Vex:xxxz _ab3/ca V * exVxz 314. Two or more radical quantities, having the same index, are said to be of the saine 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 6 two following, va?band Va%b4=a4 X6*=a”.6°= V a?b= vaľba. In like manner, the radical quuntities 2aRb and V 16a", may be reduced to other equivalent ones, having the same radical quantity ; thus, 20%b= 3¢a® X2b=a®3/26, and 16ab= Ba2.26=8. a. 26=2uV2b; where the radical factor 3/2b is common to both. 315. The addition ard subtractioo of radical quantities can in general he only indicated : Thus, yas added to, or subtracted from v b, is written vb +ya', and no farther reduction car. be made, unless we assign numeral values to a and b. But the sum of vab, vab, and ✓ 4a ́b is =avbtavb+2a v b=tavb; 3.Vabab=2 Vab; and aba+a%b4=b7atab va = b vatab va= (b tab) va, 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. 367/2a5b3, 8an 20°65, and 7ab2a2b, become, by reduction, 3ab 3, 2a*b*, Baby/2a2b>, and —7aby/2ab2 ; which are similar quantities, and their sum is =4ab2a*b*. m gn mm' anymi X 6 mm mm' mm VOP m 317. We have demonstrated, (Art. 313), this formula, my apbec"="/AP X"/69 X CT ; 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 2 요 sinyal by my69, or that of a" by 6" : we can bring this case to the preceding, by reducing to the same denominator, (Art. 152), the fractions L, and 4 ; and we shall have"/aP X69 P. pm' un apm'bqm. 319. The rule for dividing two radical quantities of the same kind, may be read in this formula (Art. 294.) ="V /b9 be and it only remains to extend it to two radical quantities of different denominations. Let therefire map be divided by m, bq: by passing from radical signs to fractional exponents, we have P "V 169 bqm* bm' b 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 VaP Xb="/ap.b! ; it becomes, by passing from radical signs to fractional expopents, P 2 Xa”=/aptiza =am m. Therefore the rule demonstrated (Art. 7!), with regard to whole positive exponents, extends to fractional exponents. 320. In the same hypothesis b=a, the quotient be pm' mm' am amm? арт! ' mm' apm' mm' another extension of the rule given (Art. 86), to fractional positive exponents. 321. We may, in the preceding formula, suppose p=0; and P 2 it becomes, (since a"="=a•=1) a transformation == m a am 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, 1 9 P 1 P m a P m m m am and if we multiply them member by member, we shall have the equal products, _1 p_2 1 1 1 X ; or a Xa =0 1 P P din It appears therefore evident, that exponentials, with fractional negative exponents, follow the same rule in their multiplication, as those with whole positive exponents. P 323. The division of am, by am, gives for the quotient m mm a 9 1 Now the exponent of the quotient, namely that 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 1 irrational exponents : For instance av2, 63, &c. since that the roots of ✓2 and v 3 might be obtained with a sufficient degree of approximationi, and such that the error may te 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 |