To simplify the expression ((root(256, 10))^5)/(512)^{(1/3)}, let's simplify each component separately.
First, we need to find the value of the radicand root(256, 10).
Since 256 is a perfect fourth power of 4 (4^4 = 256), we can rewrite root(256, 10) as the fourth root of 10 multiplied by the fourth root of 10 multiplied by the fourth root of 10 multiplied by the fourth root of 10.
So, root(256, 10) = (10^(1/4))*(10^(1/4))*(10^(1/4))*(10^(1/4)) = 10^(1/4 + 1/4 + 1/4 + 1/4) = 10^(4/4) = 10^1 = 10.
Now let's simplify (512)^(1/3):
The cube root of 512 is 8 because 8^3 = 512.
Now, we substitute these simplifications back into our original expression:
((root(256, 10))^5)/(512)^(1/3) = (10^5)/(8) = 100,000/8 = 12,500.
Therefore, ((root(256, 10))^5)/(512)^(1/3) is equivalent to 12,500.