29. The factors of a product may be taken in any order. It is proved in Arithmetic that when one number, whether integral or fractional, is multiplied by a second, the result is the same as when the second is multiplied by the first. The proof is as follows: when the numbers are integers, a and b suppose, write down a series of rows of dots, putting a dots in each row; and take b rows, writing the dots under one another as in the following scheme : a in a row b rows. Then the whole number of the dots is a repeated b times, that is a x b. Now consider the columns instead of the rows: there are clearly b dots in each column, and there are a columns; thus the whole number of dots is b repeated a times, that is b xa. Hence, when a and b are integers, abba. When the numbers are fractions, for example and, we prove as in Art. 28 that above proof for integers, 5 3 5 x 3 4 7 x 4' 7 And, by the 5 3 3 5 ; hence X == 74 47 Hence we have abba, for all positive values of a and b; and the proposition being true for any positive values of a and b, it must be true for all values, whether positive or negative; for from the preceding Article the absolute value of the product is independent of the signs, and the sign of the product is independent of the order of the factors. Hence for all values of a and b we have If in the above scheme we put c in place of each of the dots; the whole number of the c's will be ab; also the number of c's in the first row will be a, and this is repeated b times. Hence, when a and b are integers, c repeated ab times gives the same result as c repeated a times and this repeated b times. So that to multiply by any two whole numbers in succession gives the same result as to multiply at once by their product; and the proposition can, as before, be then proved to be true without restriction to whole numbers or to positive values. Thus, for all values a, b and c, we have of By continued application of (i) and (ii) it is easy to shew that the factors of a product may be taken in any order, however many factors there may be. Thus 30. Since the factors of a product may be taken in any order, we are able to simplify many products. For example: 3a x 4a= 3 x 4 x axa = 12a2, (−3a) × (-4b) = + 3a × 4b = 3 x 4 × a × b = 12 ab, (√2a)=√2α × √√/2a = √√2 × √2 × aa = 2a2. Although the order of the factors in a product is indifferent, a factor expressed in figures is always put first, and the letters are usually arranged in alphabetical order. 31. Since a2 = aa, and a3 = aaa; we have a2 × a3 = aα × aaa= a = a2+3 In the above examples we see that the index of the product of two powers of the same letter is equal to the sum of the indices of the factors. We can prove in the following S. A. 2 manner that this is true whenever the indices are positive integers: since by definition :. am × a”=(aaa...to m factors) × (aaa... to n factors) The law expressed in (D) is called the Index Law. 32. Since (-a) × (− a) = + a2 = (+ a) (+ a) [Art. 28], it follows conversely that the square root of a" is either +a ora: this is written a2=a, the double sign being read 'plus or minus.' Thus there are two square roots of any algebraical quantity, which are equal in absolute magnitude but opposite in sign. EXAMPLES. 1. Multiply 2a by -4b, a2 by - a3 and -2a3b by -3ab3. Ans. -8ab, - a3, 6a4b4. 2. Multiply -2xy2 by - 3y2z, 3ax2y by -5a2xy2, and 3a2bc2x by 12ab2cx3. Ans. 6xyz, -15a3x3y3, 36a3b3ç3x. 3. Multiply 7a4b3c2 by -3a3b5c7, and -2ab3x5y2 by -4a3b2x4y6. Ans. -21a7b8c9, 8a4b5x9y3. 4. Find the values of (− a)2, ( − a)3, ( − a) and ( − a)5. Ans. a2, -a3, aa, – a3. 5. Find the values of (ab)2, (a2b) and ( − 3ab2c3)3. Ans. a2b2, a8b4, -27a3bc9. 6. Shew that the successive powers of a negative quantity are alternately positive and negative. 7. Find the cubes of 2a2b, - 3ab2c3, and of - 2a2bx3y3. Ans. 8a6b3, -27a3b6c9 and -8a*b3x3y¤. 8. Find the values of (-a)2 × (-b)3, of (− 2ab2)3 × ( − 3a2b)3, and of (-3abc)2 × (2a2b)3. Ans. -a2b3, 216a9b9, 72a8b5c2. 9. Find the value of 3abc-2a2bc3+4c4, when a=2, b=-1, and c = −2. Ans. 12. 10. Find the value of 2a2bc-3b2cd+4c2da-5d2ab, when a= -1, b=2, c=-3 and d= -4. Ans. 148. DIVISION. 33. Division is the inverse operation to that of multiplication; so that to divide a by b is to find a quantity c such that cx b=a. Since division is the inverse of multiplication and multiplications can be performed in any order [Art. 29], it follows that successive divisions can be performed in any order. Thus a÷b÷c=a÷c÷b. It also follows from Art. 29 that to divide by two quantities in succession gives the same result as to divide at once by their product. Thus a÷b÷c=a÷ (bc), which is usually written a ÷ bc. Not only may a succession of divisions be performed in any order, but divisions and multiplications together may be performed in any order. For example For ax b÷c=a÷c × b. a = a ÷ cxc; =a÷cxbxc; [by Art. 29] therefore, dividing each by c, we have ax b÷c=a÷c × b. Hence we get the same result whether we divide the product of a and b by c, or divide a by c and then multiply by b, or divide b by c and then multiply by a. 34. The operation of division is often indicated by placing the dividend over the divisor with a line between means a ÷ b. Sometimes a/b is written for α When a ÷b is written in the fractional form, a is called the numerator, and b the denominator. so that to divide by any quantity c is the same as to multi in which form it is seen to be included in Art. 29 (C). 35. Since a3 × a2 = a3, and a2 × a3 = a10; we have conversely a ÷ a3 = a3, and a1o÷ a3 = a3. And, in general, and m>n, we have for by Art. 31 10 when m and n are any positive integers am ÷ an = am-n, am-n xan = am. Hence if one power of any quantity be divided by a lower power of the same quantity, the index of the quotient is equal to the difference of the indices of the dividend and the divisor. |