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               
      
      

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

   3. 8 ÷ 4 x 3 + 1 - (4⁰ - 1)       

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