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  Section 5-2:  Rational Exponents     All rules presented in the previous section were defined for integers only.  All of the properties in the last section can also be extended to include rational exponents according to the following definitions:  

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Examples   1) 
 
2) 
 
3) 
 
4) 
 
5) 
 
6) 
 
7)  (100^1/2 - 36^1/2)² = (10 - 6)² = 4² = 16
 
8)  x^1/2(x^3/2 + 2x^1/2) = x² + 2x
 
9) 

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We can use these rules to solve for x when x is the exponent.  This method will only work if the bases are the same.  Check back in section 5-1 for the appropriate rule!!   Examples   1)  16^x = 2⁵   We can write both sides in base two.
     2^4x = 2⁵   Now use the fact that the bases are the same, the exponents are =
      4x = 5       And solve for x!!
        x = 5/4     To check it, take the 5/4 root of 16 = 2⁵!!
 
2)  27^1-x = (1/9)^3-x   You need to make both bases the same.  How about 3!!
      3^3(1-x) = 3^-2(3-x)      Notice the power on the right side is negative.
       3(1 - x) = -2(3 - x)    Because the bases are =, the exponents must be =
        3 - 3x = -6 + 2x        Solve for x.
        3 = -6 + 5x
        9 = 5x
        9/5 = x

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 The growth and decay formula can also be used with rational numbers.  Consider the following:   1)  The cost of a computer has been increasing at 7% per year.  If it costs $1500 now, find the cost:
                     a)  2 years and 6 months from now
                     b)  3 years and 3 months ago.
             Solutions:
                     a)  A_o = 1500, r = .07 and t = 2.5
                           A(2.5) = 1500(1 + .07)^2.5 = 1500(1.07)^2.5 = 1776.44
 
                     b)  A_o = 1500, r = .07, and t = -3.25
                             A(-3.25) = 1500(1.07)^-3.25 = 1203.91

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  Let's dribble into the next section!!    I better bounce back to the previous section!