Exponentials and Order of Operations

Definition of an Exponential

Examples  

        32  =  (3)(3),       43  =  (4)(4)(4)



Definition of an Exponential  

          xn = (x)(x)(x)(x)(x)...(x)

where there are n x's.  x is called the base and n is called the exponent

 

In the first example above, 3 is the base and 2 is the exponent.  In the second example, 4 is the base and 3 is the exponent.   

 

Exercises:   Evaluate

  1. 23          8

  2. 34          81

Order of Operations

We read the following symbols in order:

  1. ( ) and | |

  2. Exponents

  3. X and /

  4. + and -

  5. If there is a tie we read from left to right.


Example

        7. (-3 + 1)2 -3 + 2.5        First work inside the parentheses

        = 7.(-2)2 - 3 + 2.5             Now evaluate the exponent

        = 7. 4 - 3 + 2.5               Next do the first multiplication

        = 28 - 3 + 2.5                Now multiply

        = 28 - 3 + 10                 Subtract

        = 25 + 10                      Finally add

        = 35

Exercises:

  1. (2 + 22 x 3 )/ 7        2

  2. 7  + (3 + 2)/3          26/3

  3. |3- 6|2/(-3)              -1

  4. (-1/2 + 1/3)(6)        -1

     

  5. -1/2 + 1/3 . 6          3/2

  6. Explain why -12 and (-1)2 are -1 and 1 respectively.
    The first squares and then makes negative, the second makes negative and then squares.                               

Evaluating Algebraic Expressions

Example:

Consider  

        x2 + 3xy + 2x

What do we get if 

        x = 1      and      y = -2?

Solution:

        (1)2 + 3(1)(-2) + 2(1)  =  1 - 6 + 2  =  -5 + 2  =  -3

Exercises:

Evaluate 

               |2x - y|  
                             
               |x2 - y2|

  1. When     x = 1,    y = 2

  2. When     x = 3,    y = 6

  3. When     x = 2,    y = 2

Application

Beginning at an elevation of 6,500 feet, you take a chair that ascends at a rate of 100 ft/min for 8 minutes.  Then you ski down a run descending at a rate of 200 ft/min for three minutes and take a different chair lift that ascends at a rate of 50 ft/minute for 5 minutes.  What is you elevation now?

Solution

        6500 + 8 . 100 +  3 . (-200) + 5 . 50

        = 6500 + 800 +  3 . (-200) + 5 . 50

        = 6500 + 800 +  -600 + 5 . 50

        = 6500 + 800 +  -600 + 250

        = 7300 +  -600 + 250

        = 6700 + 250

        = 6950

You are now at an elevation of 6,950 feet.

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