A8 APPENDIX Review CAUTION Be careful with minus signs and exponents. − =− ⋅ =− 2 1 2 16 4 4 whereas () ()()()() − = − − − − = 2 2 2 2 2 16 4 j DEFINITION an If a is a real number and n is a positive integer, then the symbol an represents the product of n factors of a . That is, … a a a a n = ⋅ ⋅ ⋅ nfactors (1) DEFINITION a0 If a 0, ≠ then a 1 0 = DEFINITION a n− If a 0 ≠ and if n is a positive integer, then a a 1 n n = − In the definition it is understood that a a. 1 = Furthermore, a a a, 2 = ⋅ a a a a, 3 = ⋅ ⋅ and so on. In the expression a ,n a is called the base and n is called the exponent , or power . We read an as “ a raised to the power n ” or as “ a to the n th power.” We usually read a2 as “ a squared” and a3 as “ a cubed.” In working with exponents, the operation of raising to a power is performed before any other operation. As examples, 43 4936 2 3 4913 2 16 53 24 5924 458 53 2 2 2 4 2 ⋅ = ⋅ = + = + = − = − ⋅ + ⋅ = ⋅ + ⋅ = + = Parentheses are used to indicate operations to be performed first. For example, 2 2 2 2 2 16 2 3 5 25 4 2 2 ( ) ( )( )( )( ) ( ) − = − − − − = + = = Table 1 illustrates why a to the power 0 equals 1 and why a negative exponent results in a reciprocal. Whenever you encounter a negative exponent, think “reciprocal.” Evaluating Expressions Containing Negative Exponents (a) 2 1 2 1 8 3 3 = = − (b) x x 1 4 4 = − (c) 1 5 1 1 5 1 1 25 25 2 2 ( ) ( ) = = = − EXAMPLE 10 Now Work problems 87 and 107 The following properties, called the Laws of Exponents , can be proved using the preceding definitions. In the list, a and b are real numbers, and m and n are integers. THEOREM Laws of Exponents a a a a a ab a b a a a a a a b a b b 1 if 0 if 0 m n m n m n mn n n n m n m n n m n n n ( ) ( ) ( ) = = = = = ≠ = ≠ + − − n 3n 3 = 3 27 3 2 = 3 9 2 1 = 3 3 1 0 = 3 1 0 −1 = −3 1 3 1 −2 = = −3 1 3 1 9 2 2 −3 = = −3 1 3 1 27 3 3 Table 1 Divide by 3. Divide by 3. Divide by 3. Divide by 3.
RkJQdWJsaXNoZXIy NjM5ODQ=