Let C(x)=

^{3}√x and F(x)=

^{4}√x. Determine the value of each of the following expressions:

(i) C(25 + C(6 + C(5 + C(25 + C(6 + C(5+.....))))))

(ii) F(14 + F(5 + F(79 + F(14 + F(5+ F(79+ .....))))))

*Can you come up with an analytic (apart from a computer program) solution?*

(In reply to

For the simple case by Gamer)

Using the trick makes this problem equivalent to solving:

(((x^3-25)^3-6)^3-5)^3-x = 0

(((x^4-15)^4-5)^4-79)^3-x = 0

If we were expected to find a rational value for each case, then this would be doable with minimal computer assistance. First we would only need the leading coefficient and constant term of each polynomial. Then a finite set of candidates would be generated by the possible fractions made by factor of the constant divided by a factor of the leading coefficient.

In both cases the leading coefficient is 1, so the only rational solutions are integer solutions.

The first equation does have an integer solution, x=3. The second does not have an integer solution. I numerically calculated 1.8918763177 as a root.