x and y are positive real numbers that satisfy logxy+logyx=174 and xy=2883√. If x+y=a+bc√, where a, b and c are positive integers and c is not divisible by the square of any prime, what is the value of a+b+c?

Answer 1

Having posted the same problem three times and not
getting a response should have given you a hint, that there might be something wrong with the question, or the way you typed it.
e.g. 2883√ makes no sense, the √ is not written behind the number
which makes x+y = a + bc√ completely ambiguous.

I will start it off:

logxy+logyx=174
the way you typed it .....

2 log xy = 174
log xy = 72
xy = 10^72 ---- huge!!
but assuming xy = √2883
so √2883 = 10^72 ??? , not likely

I have a feeling you meant:

log_xy + log_yx = 174

Answer 2

logxy + logyx = 17/4

Answer 3

log y base x + log x base y =17/4

sorry for the typo.