Question: Solve for b. 155.96=[(7.17b+77.11)^(5/3)]/[(b+25.85)^(2/3)]
If you wish to do this manually, then first of all cross-multiply to get
155.96*(b+25.85)^(2/3) = (7.17b+77.11)^(5/3) -- now cube both sides.
155.96(^3)*(b+25.85)^(2) = (7.17b+77.11)^(5) -- expand both bracketed terms, multiply out and rearrange.
-18949.41374*b^5 - 1.018960456*10^6*b^4 - 2.191688725*10^7*b^3 - 2.319123858*10^8*b^2 - 1.071329553*10^9*b - 1.91273981*10^8 = 0
or
g(b) = -18949.41374*b^5 - 1.018960456*10^6*b^4 - 2.191688725*10^7*b^3 - 2.319123858*10^8*b^2 - 1.071329553*10^9*b - 1.91273981*10^8 = 0
Now use the Newton-Raphson method to solve for b. The formula is,
b_(n+1) = b_n - g(b_n)/g'(b_n)
where b_n is the nth iterated value for b, and b_1 is the inital (guessed) starting value
Use a starting value of b_1 = -0.2 (You could get this by graphing the function g(b))
A few iterations shoud give you a result of about b = -0.1858886992