Exponents and Order of Operations   Exponents Recall that multiplication is defined as repeated addition, for example         4 + 4 + 4 + 4 + 4 + 4 + 4  =  4 x 7 What about repeated multiplication?  For example, is there an easy way to write          4 x 4 x 4 x 4 x 4 x 4 x 4 Fortunately, mathematicians have developed a convenient way of writing this.  We write         4 x 4 x 4 x 4 x 4 x 4 x 4  =  47  Here, the number 4 is called the base and the number 7 is called the exponent. We read this as four to the seventh power. Examples   Find the value of each expression 24 33 010 92 1221 109 Solutions 24  =  2 x 2 x 2 x 2  =  4 x 4  =  16 33  =  3 x 3 x3  =  9 x 3  =  27 010  =  0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0  =  0 92  =  9 x 9  =  81 1221  =  122        Notice that the 1 means the 122 only appears one time 109  =  10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10  =  1,000,000,000 For any whole number b other than zero         b0  =  1 Example         160  =  1 Order of Operations Example Consider the following expression         12 - 32 + 2 x (5 + 1) In what order should we work out this? The general rule for working out arithmetic problems is that we work them out in the following order: Inside parenthesis Exponents x and ÷  + and - If there are two operations of equal priority, we work them out from right to left. Example (Continued)         12 - 32 + 2 x (5 + 1)  =  12 - 32 + 2 x 6  Parentheses first:  5 + 1  =  6                                           =  12 - 9 + 2 x 6        Exponents next:  32  =  9                                           =  12 - 9 + 12        Multiplication next:  2 x 6 =  12                                           =  3 + 12            "-" and "+" are same so left first:  12 - 9  =  3                                           =  15 ExercisesCalculate the following  (To check your answer hold mouse over the yellow rectangle)   (2 x 3)2 - 5 + 3 x 22                4 + 6 ÷ 2 x 5 - (32 + 1)        8 ÷ 4 x 3 + 1 - (40 - 1)          Back to the Arithmetic of Whole Numbers page Back to the Math 187A page Back to the Math Department page e-mail Questions and Suggestions