Mignosi, Restivo, and Salemi (1998) proved that for all $ε> 0$ there exists an integer $N$ such that all prefixes of the Fibonacci word of length $\geq N$ contain a suffix of exponent $α^2-ε$, where $α= (1+\sqrt{5})/2$ is the golden ratio. In this note we show how to prove an explicit version of this theorem with tools from automata theory and logic. Along the way we gain a better understanding of the repetitive structure of the Fibonacci word.