5.2 The Integers 231 The number 5 to the third power, or 5 cubed, written 5 ,3 means = ⋅ ⋅ 5 5 5 5 3 3 factors of5 In general, the number b to the nth power, written b ,n means = ⋅ ⋅ ⋅ ⋅ b b b b b n n factors of b Example 9 Evaluating the Power of a Number Evaluate. a) 52 b) −( 3)2 c) 34 d) 11000 e) 10001 Solution a) = ⋅ = 5 5 5 25 2 b) − = − ⋅ − = ( 3) ( 3) ( 3) 9 2 c) = ⋅ ⋅ ⋅ = 3 3 3 3 3 81 4 d) = 1 1 1000 The number 1 multiplied by itself any number of times equals 1. e) = 1000 1000 1 Any number with an exponent of 1 equals the number itself. 7 Now try Exercise 43 Timely Tip When an expression involves a negative number and an exponent, we have to be careful when evaluating the expression. Let’s evaluate the two expressions −( 7)2 and −7 .2 The location of the negative sign is very important. In general, x x x x x ( ) ( )( )( ) ( ), n − = − − − … − and − = − ⋅ x x 1 . n n Therefore, − = − − = − = − ⋅ = − ⋅ = − ( 7) ( 7)( 7) 49 7 1 7 1 49 49. 2 2 2 n factors of −x ( ) In general, −xn means take the opposite of x ,n or − ⋅ x 1 .n For example, −52 means take the opposite of 52 or − ⋅ 1 52 to obtain an answer of 25. − Example 10 The Importance of Parentheses Evaluate. a) −( 2)4 b) −24 c) −( 2)5 d) −25 Solution a) − =−−−−=−−=−−= ( 2) ( 2)( 2)( 2)( 2) 4( 2)( 2) 8( 2) 16 4 b) −24 means take the opposite of 2 or 1 2 . 4 4 − ⋅ − ⋅ = − ⋅ ⋅ ⋅ ⋅ = − ⋅ = − 12 12222 116 16 4 c) − =−−−−−=−−−=−−− = − = − ( 2) ( 2)( 2)( 2)( 2)( 2) 4( 2)( 2)( 2) 8( 2)( 2) 16( 2) 32 5 d) − = − ⋅ =−⋅⋅⋅⋅⋅ =−⋅ = − 2 12 122222 132 32 5 5 7 Now try Exercise 45 From Example 10, we can see that − ≠ − x x ( )n n when n is an even natural number and that − = − x x ( )n n when n is an odd natural number. Order of Operations Now that we have introduced exponents, we can present the order of operations . Can you evaluate the expression + ⋅ 2 5 4? Is it 28? Or is it 22? To answer this, you must know the order in which to perform the indicated operations.
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