Since, by the Commutative Law, 2n÷ 2 = 2 ÷ 2 x n = 1 x n = n, 2 n is always an even number when ʼn is an integer. EXERCISES X. Find, in the most convenient way, the values of: 1. 25 x-12 ÷ +5. 2. -100 x-7÷-25. 4. +331-20 ×+3. 5. -30-9 x-12. 3. -1000 x-11 ÷ +125. 6. 10+17 x −34. Find, in the most convenient way, the value of a ÷ b÷c × d, 7. When a = +170, b = −3, c = +17, d = −6. Find the values of the following expressions, first removing the parentheses: 9. +25 × (+12 ÷-4). 10. 20÷(−5÷+2). 12. -600÷(-200÷−25×+3÷-4). 13. +300 11. +100÷(+25 ×−2). (-150÷+6x+8÷−4). § 5. ONE SET OF SIGNS FOR QUALITY AND OPERATION. 1. In conformity with the usage of most text-books of Algebra we shall in subsequent work use the one set of signs, + and, to denote both quality and operation. For the sake of brevity the sign is usually omitted when it denotes quality; the sign is never omitted. Thus, instead of +2, we shall write +2, or 2; instead of -2, we shall write - 2. 2. We have used the double set of signs hitherto in order to emphasize the difference between quality and operation. It should be kept clearly in mind that the same distinction still exists. We now have N++2=N+(+2) = N+ 2, omitting the sign of quality, +; N+2=N+(-2), wherein + denotes operation, and - denotes quality. N-+2=N-(+2) = N − 2, omitting the sign of quality, +; N--2-N-(-2), wherein the first sign, -, denotes operation, the second sign, -, denotes quality. 3. In the chain of operations (+2) + (− 5) − (+ 2) − (− 11) the signs within the parentheses denote quality, those without denote operation. That expression reduces to (+ 2) − (+ 5) − (+ 2) + (+11), or 25-2+11, dropping the sign of quality, +. In the latter expression all the signs denote operation, and the numbers are all positive. 4. The following examples illustrate the double use of the signs + and Ex. 1. +4++3 = + 4 + (+ 3) = 4+3=7. Ex. 2. −5++2 = − 5 + (+ 2) = − 5 + 2 = −3. Ex. 3. +7 —-5=+7 − (− 5) = + 7 + (+ 5) = 7 + 5 = 12. " Ex. 4. -4x+3= − 4 × (+ 3) = −4 × 3 = — 12. Ex. 5. -4x-3=-4x (-3)= 12. EXERCISES XI. Find the values of the expressions in Exx. 1-8, first changing them into equivalent expressions in which there is only the one set of signs, + and -: Find the values of the expressions in Exx. 9-14, first changing them into equivalent expressions in which there is only one set of signs,+ and and then removing the parentheses: Find the values of the expressions in Exx. 15-22, first changing them into equivalent expressions in which there is only one set of signs, + and -: = 28. When a = — 5, b — 3, c = 4, d = − 5, e = − 7 ? 29. When a = 12, b: =- - 2, c=- - 9, d 13, e = 28 ? Find the results of the following indicated operations: § 6. POSITIVE INTEGRAL POWERS. 1. A continued product of equal factors is called a Power of that factor. Thus, 2 × 2 is called the second power of 2, or 2 raised to the second power; aaa is called the third power of a, or a raised to the third power. In general, aaa ... to n factors is called the nth or a raised to the nth power. of power a, The second power of a is often called the square of a, or a squared; and the third power of a the cube of a, or a cubed. 2. The notation for powers is abbreviated as follows: a2 is written instead of aa; a3 instead of aaa; 3. The Base of a power is the number which is repeated as a factor. E.g., a is the base of a2, a3, ..., a". The Exponent of a power is the number which indicates how many times the base is used as a factor, and is written to the right and a little above the base. E.g., the exponent of a2 is 2, of a3 is 3, of a" is n. The exponent 1 is usually omitted. Thus, a1= a. An exponent must not be confused with a subscript. Thus, a3 stands for the product aaa; while a, is a notation for a single number. 4. The definition of a power given above requires the exponent to be a positive integer. In a subsequent chapter this definition will be extended to include powers with negative and fractional exponents. Notice that the words positive integral refer to the exponent and not to the value of the power, which may be negative or fractional. E.g., (-2)=(-2)(2)(-2)=-8; (3)2 = 4 × } = 1. 5. The base of a power must be inclosed within parentheses to prevent ambiguity: (i.) When the base is a negative number. Thus, (— 5)2 = (— 5)( — 5)=25; while -52 — — (5 × 5) — — 25. == (ii.) When the base is a product or a quotient. Thus, (2 × 5)3 = (2 × 5) (2 × 5) (2 × 5) = 1000; 2 × 53 = 2(5 × 5 × 5) = 250. while (iv.) When the base is itself a power. Thus, (23)2 = 23 × 23 = (2 × 2 × 2) (2 × 2 × 2) = 64, while 23* = 23×3 = 2o = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512. Express the following powers in the abbreviated notation: 23. (a + x)(a + x) (a + x) (a + x). 24. (aaa - b)(aaa — b)(aaa — b) Express in algebraic notation : ... to 17 factors. 25. The sum of the squares of a and b. 28. The cube of the sum of x, y, and z. Properties of Positive Integral Powers. 6. (i.) All (even and odd) powers of positive bases are positive. E.g., 232 x 2 x 2 = 8. 3+= 3 x 3 x 3 x 3 = 81. In general, (+ a)" = (+ a) (+ a) (+ a) ... n factors. =+a”, n even or odd. (ii.) Even powers of negative bases are positive; odd powers of negative bases are negative. Notice that the words even and odd refer to the exponents. Also, that, for all integral values of n, 2n is even (§ 4, Art. 9), and hence that 2n+1 or 2 n -1 is odd. E.g., (−2)* = (-2) (−2) (-2) (-2) = 16; (— 5)3 = (— 5) (-5) (-5)=-125. |