Find the required digit What is the digit at the $50^{th}$ place from left of $(\sqrt50 +7)^{50}$  

I thought of binomial expansion but it was way too lengthy. Can anyone suggest any other  way?


Consider the number $N=(7+\sqrt{50})^{50} +(7-\sqrt{50})^{50}$. By looking at the binomial expansion of each term, or in several other ways, one can verify that $N$ is an integer. 
The number $7-\sqrt{50}$ has absolute value about $0.071$. The $50$-th power is well under $10^{-50}$, indeed less than $4\times 10^{-58}$. 
Thus the number we were given, plus a positive number well under $10^{-50}$, is the integer $N$. It follows that the $50$-th digit after the decimal point is $9$. 
Remark: Whenever $7+\sqrt{50}$ has a problem, its friend (conjugate) $7-\sqrt{50}$ is ready to help. 
