POSITIVE INTEGRAL POWERS. 32. The Sign of Continuation, is read, and so on, or and so on to; as 1, 2, 3, ......, read, one, two, three, and so on; or 1, 2, 3, ....., 10, read, one, two, three, and so on to 10. 33. 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 power of a, or a raised to the nth power. 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. 34. The notation for powers is abbreviated as follows: a2 is written instead of aa; as instead of aaa; 35. The Base of a power is the number which is repeated as a factor. E.g., a is the base of a2, a3, ......, a”. 36. 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. 37. 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, while (2+3)2=(2+3) (2+3)= 5 x 5 = 25; 2+32=2+3x3=2+9= 11. (iv.) When the base is itself a power. Thus, (28)228 x 23 (2 × 2 × 2) (2 × 2 × 2) = 64. = while 232 23×3 = 29 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512. EXERCISES VIII. Express each of the following powers in the abbreviated notation: 4. (− a) (— a). 7. - nn nnn. 10. (a+b) (a + b) (a + b). 11. (x — yy) (x — yy) (x — yy). 12. (a+b) (a+b) (a + b)... to 12 factors. Express each of the following powers as a continued product: 24. The cube of the sum of a and b. 25. The length of a side of a square floor is a feet. How many square feet in the floor? 26. A field is 3 a rods long and 2 a rods wide. How many square rods in its area? 27. A box is 4 x feet long, 3 x feet wide, and 2 x feet high. How many cubic feet does it contain? Properties of Positive Integral Powers. 38. (i.) All (even and odd) powers of positive bases are positive. E.g., 22x2x2 = 8. 34 = 3 × 3 × 3 × 3: = 81. (ii.) Even powers of negative bases are positive; odd powers of negative bases are negative. Find the value of each of the following powers: Find the value of each of the following expressions: 21. 22+32. 22. (2+3)2. 23. 33 — 23. 25. (4 × 3)2. 26. 6 × 42. 27. 2(-3)3. 24. (3-2)3. 28. [2(-3)]3. -4, c=2, find the value of each of the 30. ba. 31. (ab). 32. bca. 33. (abc). 36. (a2b2+c2)2. CHAPTER III. THE FUNDAMENTAL OPERATIONS WITH INTEGRAL ALGEBRAIC EXPRESSIONS. DEFINITIONS. 1. An Integral Algebraic Expression is an expression in which the literal numbers are connected only by one or more of the symbols of operation, +, -, x, but not by the symbol÷. E.g., 1 + x + x2, 5a2bcd2, etc. 2. The word integral refers only to the literal parts of the expression. E.g., ab is algebraically integral; but when a = 1, b = 3, we have 3. Coefficients. — In a product, any factor, or product of factors, is called the Coefficient of the product of the remaining factors. E.g., in 3 abc, 3 is the coefficient of abc, 3b of ac, etc. A Numerical Coefficient is a coefficient expressed in figures. E.g., in 3 ab, - 3 is the numerical coefficient of ab. A Literal Coefficient is a coefficient expressed in letters, or in letters and figures. E.g., in 3 ab, a is the literal coefficient of 3b, and 3 a of b. The coefficients +1 and -1 are usually omitted. 4. A coefficient must not be confused with an exponent. E.g., 4a= a + a + a + a; while a1: = a xaxaxa. 1 5. The sign+, or the sign, preceding a product, is to be regarded as the sign of its numerical coefficient. Thus +3 a means the product of positive 3 by a; -5x means the product of negative 5 by x. In particular, + a means the product of positive 1 by a, and a means the product of negative 1 by a, unless the contrary is stated. 5. If the sum, a + a + a + a, be represented as a product, what is the coefficient of a ? 6. If the algebraic sum, −b-b− b − b — b, be represented as a product, what is the coefficient of b? Of b? ... 7. If the sum ax + ax + ax + to 10 terms be represented as a product, what is the coefficient of ax? Of x? 6. Like or Similar Terms are terms which do not differ, or which differ only in their numerical coefficients. E.g., in the expression +3 a + 6 ab − 5 a +7 ab, +3a and 5 a are like terms; so are + 6 ab and +7 ab. Unlike or Dissimilar Terms are terms which are not like. 7. A Monomial is an expression of one term; as a, 7 bc. A Binomial is an expression of two terms; as −2 a2+3 bc. A Trinomial is an expression of three terms. E.g., a+b-c, -3a2+7b3-5 ct. A Multinomial * is an expression of two or more terms, including, therefore, binomials and trinomials as particular cases. E.g., a + b2, a2 + b — c3, ab + bc — cd — ef. * The word Polynomial is frequently used instead of Multinomial. |