Let
\(P = 5^{1/5} \cdot 5^{1/25} \cdot 5^{1/125} \dotsm\)
Then P can be expressed in the form a^{b/c}, where a, b, and c are positive integers. Find the smallest possible value of a + b + c.
Using an exponent law, we can write P as
5 ^ ( 1/5 + 1/25 + 1/125 .....)
The sum of the exponents is the sum of an infinite series =
(1/5) / ( 1 - 1/5) = (1/5) / (4/5) = 1/4
So P can be expressed as
5^(1/4)
a = 5 b = 1 c = 4
a + b + c = 10
