SOLUTION: Help me: Find the Domain and solve: {{{0.04^(3-sqrt(3x+2))=25*5^sqrt(3x+2)}}}

Algebra ->  Equations -> SOLUTION: Help me: Find the Domain and solve: {{{0.04^(3-sqrt(3x+2))=25*5^sqrt(3x+2)}}}       Log On


   

Question 1183075: Help me:
Find the Domain and solve:
0.04%5E%283-sqrt%283x%2B2%29%29=25%2A5%5Esqrt%283x%2B2%29
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the Domain and solve 0.04%5E%283-sqrt%283x%2B2%29%29=25%2A5%5Esqrt%283x%2B2%29
~~~~~~~~~~~~ 

Your starting equation is

    0.04%5E%283-sqrt%283x%2B2%29%29=25%2A5%5Esqrt%283x%2B2%29.      (1)


The domain is the set of all x, such that  3x + 2 >= 0,  i.e.  x > -2%2F3.


To solve the equation, notice that  0.04 = 1%2F25;  so, equation (1) takes the form


    %281%2F25%29%5E%283-sqrt%283x%2B2%29%29 = 25%2A5%5Esqrt%283x%2B2%29,   or

    
    25%5E%28sqrt%283x%2B2%29-3%29 = 5%5E2%2A5%5Esqrt%283x%2B2%29,  which is the same as


    5%5E%282%2A%28sqrt%283x%2B2%29-3%29%29 = 5%5E%28sqrt%283x%2B2%29%2B2%29.


The bases are the same in both sides of the equation;  hence, the indexes are the same


    2%2A%28sqrt%283x%2B2%29-3%29 = sqrt%283x%2B2%29%2B2


which implies


    sqrt(3x+2) = 2 + 2*3 = 8

    3x+2 = 8^2 = 64

    3x = 64 - 2 = 62

     x = 62%2F3.      ANSWER

Solved.