DIVISION OF FRACTIONS. 46. RULE.-Invert the divisor and proceed as in Multiplication. 20 d 47. Miscellaneous Examples in the operations performed in Algebraic Fractions DIVISION OF FRACTIONS. 46. RULE.-Invert the divisor and proceed as in Multiplication. +7. Miscellaneous Examples in the operations performed in Algebraic Fractions (5) c+2ab—3uc— a+b a d = 56 bey 16 abc 15 cdf-4deg 24 b c 4 6 efg (e-f) — 3 g + + 2 ƒ +1 6 efg a— cx + dx' +' b'c — 5 ab' c+a' 2 ab-bc'+3abc-a br — b c b'-bc 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. 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)2 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 √ 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. 2 2 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, (5 a 2 b3 c) 2 So, 3 = 5a2b3c × 5 a 2 b 3 c = 25 a b c 2 m 2 3 4 6 =9ab2c3 d 4 × 9 a b2 c3 d4 81 a 2 b c d 2 (A x y z 2 n - · - -)2=Axm y "z"---X Ax" y " z " - - - = A 2 x 2 my 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 Arith metic. 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 48 is not a square number, and the exponent of a is not an even number. In this case we introduce the quantity into our calculations, affected with the sign |