Given that 3^x + 4^x =5^x. Solve for x

Answer 1

could be a very nasty equation
but ....
3^x + 4^x = 5^x
by inspection and recall, I know
that 3^2 + 4^2 = 5^2
(did you recognize the 3-4-5 right-angled triangle)

so x = 2

now suppose it had been
3^x + 4^x = 6^x
look at Wolfram's solution
http://www.wolframalpha.com/input/?i=3%5Ex+%2B+4%5Ex+%3D6%5Ex
x = 1.29317

Now, how do you think they got that ?