Exponents and Order of Operations

 

  1. 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

    1. 24

    2. 33

    3. 010

    4. 92

    5. 1221

    6. 109

    Solutions

    1. 24  =  2 x 2 x 2 x 2  =  4 x 4  =  16

    2. 33  =  3 x 3 x3  =  9 x 3  =  27

    3. 010  =  0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0  =  0

    4. 92  =  9 x 9  =  81

    5. 1221  =  122        Notice that the 1 means the 122 only appears one time

    6. 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

  2. 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:

    1. Inside parenthesis

    2. Exponents

    3. x and  

    4. + 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

    Exercises

    Calculate the following  (To check your answer hold mouse over the yellow rectangle)

     

    1. (2 x 3)2 - 5 + 3 x 22                43

    2. 4 + 6  2 x 5 - (32 + 1)        9

    3. 8  4 x 3 + 1 - (40 - 1)        7

 



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