show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function：http://en.wikipedia.org/wiki/Gamma_function I found this problem is American Mathematical Monthly (1975,E5952 problem) and post by J.Gilles, and I found this problem can't solution,and I think this is very interesting problem,
How find this problem solution? Thank you everyone
On
It's surprising the problem made it into the American Mathematical Monthly.
There is a known and conjecturally complete set of situations where a product of powers of $\Gamma(x)$ at several rational values of $x$, is algebraic. $\Gamma(8/9)$ is not one of them. Allowing powers of $\pi$, there is no way to isolate an individual value of $\Gamma(a/b)$ with $b>2$ using the functional equations, and a simple formula for such a number is very hard to believe.
You can find an abstract description of the algebraic gammas recipe in the considerably more abstract article by Pierre Deligne on L-functions and periods of integrals (or the appendix to that article, written by Nicholas Katz). This was published later than the Monthly problem, but the part about algebraic $\Gamma$ products was widely understood informally, and not only to experts, for a very long time prior to its formal statement in Deligne's article.
Problem 5953 (not E) ... Monthly 1974 p 1033 correction...
Solution ... Monthly 1975 p 679