Ex. 1. (2 × 5)2 = (2 × 5)(2 × 5) = (10)(10) = 100. The example above should be distinguished from the following: Ex. 2. Ex. 3. 2 × 52 = 2(5 × 5) = 2(25) = 50. (3) 2 3 3 9 = X = If the numerator alone or the denominator alone is to be raised to a power, we may write 32 42 9416 Similarly, 16 (iii.) The base a sum or a difference. Ex. 4. (3 + 5)2 = (3 + 5)(3 + 5) = (8)(8) = 64. This should be distinguished from the following: = = Ex. 6. (32)8 (32)(32)(32) = (3 × 3)(3 × 3)(3 × 3) = 9 × 9 × 9=729. The use of exponents above should be distinguished from 323, which may be taken to mean either (32)3 (read "the cube of the second power of 3") 3(28) or (read "3 raised to the power two cubed "). Using the letters a, b, c, x, y, z, etc., to represent positive whole numbers, find expressions for the following: Find the values of the following expressions : 49. +28++32. 52. +62 +32. 55. +25-25. 53.58 ++34. 56. +12+22++32++42. 54. 10292. 57. +12+32 + +52++72. 58. +2232+ +42 — −52. 59. (+42 — +52)2 — (−62 — −72)2. II. DIVISION 36. The terms dividend, divisor, quotient, remainder are used relatively in the same way in algebra as in arithmetic. Division as an operation is the inverse of multiplication. To divide one number (dividend) by another (divisor), is to find another number (quotient), which when multiplied by the divisor produces the first (dividend). E. g. To divide 12 by 4 is to find the quotient 3. Multiplying the quotient 3 by the divisor 4 produces the original dividend 12. 37. By the mutual relation of multiplication and division the quotient has the fundamental property that, when multiplied by the divisor, the product is the dividend. That is, Quotient × Divisor = Dividend. If we represent dividend, divisor, and quotient by D, d and Q respectively, we may indicate the quotient by writing D and our definition of division d 38. Division, like subtraction, cannot always be performed, but it may always be indicated. It is only in exceptional cases that there can be obtained an integral quotient with no remainder. In this case the dividend is said to be exactly divisible by the divisor. 39. The fractional notation for a quotient, namely,, and the solidus notation a/b, are commonly used for division. Primarily, either means that we are to take the bth part of unity a times as a summand. Hence, b times a of the 6th parts of unity is equivalent to a times unity; or in symbols, a (a ÷ b) × b = a. Hence has the same meaning as a ÷b when a and b are whole numbers. 40. When division can be performed at all, it can lead to but a single result; hence it is called a determinate process. Division by 0 is not an admissible operation. 41. Since multiplication and division are mutually inverse operations, it follows that if any number be successively multiplied by, and then divided by the same number, or be first divided by and then multiplied by the same number, the resulting value will be the same as though no operation had been performed. Or, stated in symbols, and (a ÷ b) × b = a, 42. It follows, from the definition of division, that if the product of two factors be divided by either of the factors, the resulting quotient will be the other factor. Or, and (a × b) ÷ a = b, (a × b) ÷ b = a. Since in the product (a × b) the factors a and b of the dividend are separated by the multiplication sign, it is merely a matter of inspection to obtain the second member of each identity. 43. The Law of Quality Signs for division may be obtained directly from the set of identities in § 10 by applying the definition of division. (i.) +(ab) ÷ +b = +a, (ii.) +(ab) ÷ ̄b = −a, (iii.) −(ab) ÷ +b = −a, (iv.) -(ab)÷¬b = +a. It should be observed that the quotient is positive whenever the signs of quality of the dividend and divisor are like, as in (i.) and (iv.), and the quotient is negative whenever the signs of quality of the dividend and divisor are unlike, as in (ii.) and (iii.). 44. It follows that the quotient obtained by dividing any number by 1 is equal to the number itself. It follows, also, that the quotient obtained by dividing any number by 1 is a number equal in absolute value to the dividend but opposite in quality. E. g. +5 ÷ +1 +5, -7 ÷ +1=-7, 45. The quotient obtained by dividing 1 by any number is called the reciprocal of the number. E. g. The reciprocal of 5 is . 46. Since the product of any number multiplied by its reciprocal is by definition +1, it follows that any number and its reciprocal have the same quality. E. g. The numbers -3 and are reciprocals, and both are negative -3 numbers. 47. Dividing by any number, except 0, produces the same result as multiplying by the reciprocal of that number. Representing any number by A, and any other number different from 0 by d, we may represent the product of A and the reciprocal of d by writing Ax (1÷d). If this expression be multiplied by d, the result is A. Hence, A × (1÷d) is equal to the quotient A÷d, that is, A × (1÷d) = A ÷ d. E. g. The quotient 12÷ 3 is equal to the product 12 × 3 48. As an extended definition of division, to correspond to that of multiplication, we have the following: To divide one number by another is to do to the first that which must be done to the second to obtain the positive unit +1. 49. In the division of positive and negative numbers, we may have I. The Divisor Positive Ex. 1. Divide +24 by +6. By the extended definition of division, the quotient resulting from the division of +24 by +6 may be obtained by performing upon the dividend +24 such operations as must be performed upon the divisor +6 to obtain the unit of positive numbers +1. Since +6 +1 × 6, it appears that we may obtain the unit of positive numbers from +6 by dividing the absolute value of +6 by 6. Hence the quotient of +24 ÷ +6, is a positive number obtained by divid ing the absolute value of +24 by 6; that is: +24÷ +6 Ex. 2. Divide -30 by +10. +(24 ÷ 6) = +4. Reasoning as before, the quotient will be the negative number obtained by dividing the absolute value of the dividend -30 by 10. That is, 30 +10 = −(30 ÷ 10) = −3. II. The Divisor Negative Ex. 3. Divide +32 by -16. By the extended definition of division, we may obtain the quotient resulting from the division of +32 by -16 by treating the dividend +32 in the same way as we treat the divisor -16 to obtain the unit of positive numbers +1. By first reversing the quality of -16 we may, from the positive number thus obtained, +16, obtain the unit of positive numbers +1, by dividing the absolute value of the result by 16. Hence we may obtain the desired quotient by first reversing the quality of the dividend +32, and dividing the absolute value of the result thus obtained by 16. That is, +32÷-16 (32 ÷ 16) = −2. Ex. 4. Divide -40 by -8. In order to obtain the unit of positive numbers +1 from the divisor -8, we may first reverse the quality of the divisor, obtaining a positive number +8, and then divide the absolute value of the number thus obtained by 8. Hence we may perform the same steps with respect to the dividend -40. That is, -40 −8 = +(40 ÷ 8) = +5. |