Exponential functions

Definition of an exponent

Exponents are shorthand notation for repeated multiplication. 
     a4 = a × a × a × a

Rules of exponents 


     a0 = 1
zero exponent 

           1
     a-n =   
           an
 negative exponents 

  am an = am + n
product rule (same base) 

  (ab)m = ambm
product rule (same exponent) 

   am
      = am - n
   an
quotient rule (same base) 

  æ a ö m    am
  ç   ÷   =    
  è b ø      bm
quotient rule (same exponent) 

   (am)n = amn
 power rule

Example:     Simplify (3x2)(-5x-5) 
Multiply the numerical coefficients
-15 x2 x-5
Use the product rule and add the exponents
-15 x-3
Use the definition of a negative exponent
-15
   
x3
Example:     Simplify (25x6y4)1/2 
Use the product rule (same exponent)
251/2(x6)1/2(y4)1/2
Compute the square root of 25 and use the power rule
5 x3y2

Base e

e = (1 + 1/n)n as n -> ¥
e = 2.718281828459045235

Exponential functions

Exponetial functions are easily recognized since the variable is in the exponent. 
     f(x) = bx
The graphs of exponential functions are either increasing or decreasing.
Exponential functions

Business applications

Compound interest is a problem that can be solved using repeated multiplication or exponents.

Example:     Suppose you have $100 in an account earning 5% a.p.r. compounded annually for 10 years. At the end of the first we have 5% more than we started with. To increase $100 by 5% we multiply $100 by 1.05 to obtain $105. At the end of the second year we have 5% more than we had at the end of the first year or $100(1.05)2 = $110.25. The pattern is that at the end of each year we increase the amount of money in the account by 5% by multiplying by an additional factor of 1.05. At the end of 10 years the amount of money in the bank is $100(1.05)10 = $162.89.

Exercises

(1) Use the rules of exponents to simplify x2 x5. Write your answer in terms of positive exponents. 
x10  
x7  
x3  
x2.5  

(2) Use the rules of exponents to simplify x7/x3  
x21  
x10  
x4  
x7/3  

(3) Use the rules of exponents to simplify (36x4y6)1/2. 
6x2 y3  
18x4 y6  
18x2 y3  
6x y2  

(4) The graph of f(x) = 3x could be
Exponential function graph


    






a)
b) 
c)
d) 


(5) Suppose $1,000 is placed in an account earning 6% interest
    compounded annually.  The amount of money in the account as
    as function of time is given by
A = 1000(1.6)t
A = 1000(1.06)t
A = 1000 + 0.06t
A = 1000(0.06)t

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