What minimum number (A) can be taken so that (A)^N is larger than the product of N numbers? Given a sequence of N numbers say 2,8,4,7,6,5. How can we calculate a minimum number say A such that AN is greater than the product of 2*8*4*7*6*5 = 13440?
So the minimum number satisfying the above condition is 5.
As 56 = 15625 which is greater than 13440.
But 46 = 4096 which is less than 13440.

The geometric mean will do it: in your case we have

$\sqrt[6]{2 \cdot 8 \cdot 4 \cdot 7 \cdot 6 \cdot 5} = (2 \cdot 8 \cdot 4 \cdot 7 \cdot 6 \cdot 5)^{\frac{1}{6}} = 4.87603...$ so, as you noted, $5^6 > 2 \cdot 8 \cdot 4 \cdot 7 \cdot 6 \cdot 5$ but $4^6 < 2 \cdot 8 \cdot 4 \cdot 7 \cdot 6 \cdot 5$.

In general, if we take $N$ numbers $a_1,..., a_N$ then setting $b =\sqrt[N]{a_1 \cdot a_2 \cdots a_N}$ we have 
$$b^N = a_1 \cdot a_2 \cdots a_N$$ and so $x^N > a_1 \cdot a_2 \cdots a_N$ for any $x>b$
