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 2. 34 Order of Operations

 We read the following symbols in order: ( ) and | | Exponents X and / + and - 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.            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. 7  + (3 + 2)/3 3. |3- 6|2/(-3) 4. (-1/2 + 1/3)(6) 5. -1/2 + 1/3 . 6 6. Explain why -12 and (-1)2 are -1 and 1 respectively. Evaluating Algebraic Expressions

Example:

Consider

x+ 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.