61. Let us now proceed to execute upon radicals, the fundamental operations of arithmetic. ADDITION AND SUBTRACTION OF RADICALS. DEFINITION.-Radicals are said to be similar when they have the same index, and when the quantity under the radical sign is the same in each; thus, 3√ a, 12 a c√ā, 15 bya, are similar radicals, as are also, 4 abmu p3, 5 l m n2 p3, 25 d'√ m n2 p3, &c. 2 This being premised, in order to add or subtract two similar radicals we have the following RULE Add or subtract their coefficients, and place the sum or difference, as a coffi cient, before the common radical. For example, (3.) 3 p q √ m n + 4 1 V m n = (3 p q + 4 1) √√ m n (4.) 9cd-4cd√a = 5c da If the radicals are not similar, we can only indicate the addition or subtraction by interposing the signs + or → It frequently happens, that two radicals, which do not at first appear similar, may become so by simplification; thus, (7.) /8a3b+16a *—†/b*+2ab3 = †/&a° (b+2a)—V/b°(b+za) (8.) 3 /4a2 + 2 1/2 a = (2 a—b) i/e a + b = 3 1/2a + 2 V2a MULTIPLICATION AND DIVISION OF RADICALS. 62. In the first place, with regard to radicals which have the same index, let it be required to multiply or divide a by V5, then we shall have Và x và = vào,and và : Với = For if we raise Vā× Võ, and Vab, each to the power of n, we obtain the same result ab; hence these two expressions are equal 44. When the denominators of the fractions which it is required to reduce are expressed in numbers, the result will frequently be much simplified by finding the least common multiple of the denominators, and then reducing the fractions to their least common denominator, according to the method explained in Arithmetic. Thus, if we are required to reduce the following fractions: The least common multiple of 4 and 5 is 20, the denominator of the third fraction; therefore the fractions, when reduced to their least common denominator, are the least common multiple of 3, 4, 6 is 12, which will be the least common denominator, and the above fractions become 12c+81-27x-10x-4-61+8x+20+29+4x-5+37x 24+60 12 = 12 =2x+5 MULTIPLICATION OF FRACTIONS. 45. RULE.—Multiply all the numerators together for a new numerator, and alt the denominators together for a new denominator. Thus, FORMATION OF POWERS, AND EXTRACTION OF ROOTS OF RADICALS 63. Let it be required to raise a to the nth power; then, (Va)" = Vax Va × Va · to n factors, =Va according to the rule for multiplication just established Hence we have the following RULE. In order to raise a radical quantity to any given power, raise the quantity under the sign to that power, and affect the result with the radical sign with its original index. If there be any coefficient, we must raise the coefficient separate ly, to the required power. Thus, When the index of the radical is a multiple of the exponent of the power which we wish to form, the operation may be simplified. @= Let it be required for example, to square Va, we have seen (Art. 58,) that √za=√ √za; but in order to square this quantity, it is sufficient to suppress the first radical sign, hence, (V2a)2 = √/2a. Again, let it be required to raise abc to the power of 5; now, "Vabc=√ √abc, but in 10 order to raise this quantity to the power of 5, it is sufficient to suppress the first radical sign; hence, (Vabc) = Vabc, and in general, that is to say, If the index of the radical be divisible by the index of the required power, we may divide the index of the radical by the index of the power, and leave the quantity under the sign unchanged. 64. With regard to the extraction of roots, by virtue of the principle established in (Art. 59), we shall manifestly have the following RULE In order to extract any root of a radical quantity, multiply the index of the radical by the index of the root required, and leave the quantity under the sign unchanged. If there be any coefficient, we must extract its root separately. Thus, ! ON THE FORMATION OF POWERS, AND THE EXRACTION OF ROOTS OF ALGEBRAIC QUANTITIES. 48. We begin by considering the case of monomials, and, in order to simplify the subject as much as possible, we shall first treat of the formation of the square and the extraction of the square root only, and then proceed to generalize our reasonings in such a manner as to embrace powers and roots of any degree whatsoever. 2 DEFINITION. The square root of any expression is that quantity which, when multiplied by itself, will produce the proposed expression. Thus the square root of a2 is a, because a, when multiplied by itself, produces a 2; the square root of (a + b) is a + b, because a + b, when multiplied by itself, produces (a+b)2; in like manner 8 is the square root of 64, 12 of 144, and so on. The process of finding the square root of any quantity is called the extraction of the square root. The extraction of the square root is indicated by prefixing the symbol V to the quantity whose root is required. Thus √ a signifies that the square root of a is to be extracted; √ a 2 + 2 a b + b2 signifies that the square root of a 2 + 2 a b + b2 is to be extracted, &c. In order to discover the method which we must pursue in order to extract the square root of a monomial, let us consider in what manner we form its square. According to the rule for the multiplication of monomials, =9ab2c3 d 4 × 9 a b 2 c 3 d1 81 a 2 b + c ® do (A x y z1---)2=Ax" y "z 49. Hence it appears, that, in order to square a monomial, we must square its coefficient, and multiply the exponents of each of the different letters by 2. Therefore, in order to derive the square root of a monomial from its square, we must, L Extract the square root of its coefficient according to the rules of Arithmetic. 50. It appears from the preceding rule, that a monomial cannot be the square of another monomial unless its coefficient be a square number, and the exponents of the different letters all even numbers. Thus 98 a b is not a perfect square, for 18 is not a square number, and the exponent of a is not an even number. this case we introduce the quantity into our calculations, affected with the sign In |