Examples for Practice in the Square and Cube Roots of Numbers. Ex. 1. Required the square root of 106929. 106929(327 9 62169 124 647 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. Ans. 73. Ex. 5. Required the cube root of 389017. 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 ar, 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 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 denominator its root. Thus, an is the nth root of the mth power of a, and is usually m 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 indexm 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 Var, as has been already observed, (Art. 18). Quantities having the radical sign v prefixed to them, are called radical Quantities: thus, a, vb, Vc, xm, &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, 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 two following, va3b2 and ab=a xb=a2.b2= a2b2= a2b2. In like manner, the radical quantities 2ab and 16a3, may be reduced to other equivalent ones, having the same radical quantity; thus, 22ab=z/a° x 3/2b=a2 3/26, and 16a3b=8a3.26 = 8. a3. 26 = 2a/26; where the radical factor 26 is common to both. 3 315. The addition and subtraction of radical quantities can in general be only indicated : Thus, Va added to, or subtracted from vb, is written ba2, and no farther reduction can be made, unless we assign numeral values to a and b. But the sum of Va2b, Vab, and v4a2b is =a vb+ ayb+2a vb=4a1b; 3/ab-ab=22ab; and Vab2+a*b*=bva+abz/a2=bva+ab va=(b+ αδ) να. 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 362/2a5b2, 8a2a2b5, and -7ab2a2b2, become, by reduction, 3ab2a2b2, 8ab2a262, and -7ab32a2b2; which are similar quantities, and their sum is =4ab/2a2b2. 317. We have demonstrated, (Art, 313), this formula, maphic*=m/arm/bqxmc2; 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, 9 the product of "a" by "/6s, or that of am by bm: we can bring this case to the preceding, by reducing to the same denominator, (Art. 152), the fractions, pm m and 2; and we shall have"/arx"/bq=ambm=amm2 X qm m bmm2 = mm2/apm x mm2/bqm=mm/apm'вят. 319. The rule for dividing two radical quantities of the same kind, may be read in this formula (Art. 204). and it only remains to extend it to two radical quantities of different denominations. : Let therefore a be divided by /bq: by passing from radical signs to fractional exponents, we have pm amm mm 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 m it becomes, by passing from radical signs to fractional amxam="/ar+q=am Therefore the rule demonstrated (Art. 71), with regard to whole positive exponents, extends to fractional exponents, 320. In the same hypothesis b=a, the quotient, another extension of the rule given (Art. 86), to fractional positive exponents. 321. We may, in the preceding formula, suppose P p=o; and it becomes, (since am=ama°=1) a 1 2 am, a transformation demonstrated, (Art. 86), in the case of whole exponents, and which still takes place when the exponents are fractional. 1 |