If \(a\) and \(b\) are positive ingegers, such that \(a^{b}=125\), then \((a-b)^{a+b-4}=\) ?

Given \(a^b = 125\) and knowing \(a\) and \(b\) are positive integers, let’s first identify \(a\) and \(b\).

Finding \(a\) and \(b\)

The number 125 is a perfect cube, specifically \(5^3\). Therefore, \(a = 5\) and \(b = 3\).

Calculating \((a-b)^{a+b-4}\)

Now, substitute \(a = 5\) and \(b = 3\) into the expression:

\[
(a-b)^{a+b-4} = (5-3)^{5+3-4}
\]

This simplifies to:

\[
(2)^{4} = 16
\]

Therefore, the value of \((a-b)^{a+b-4}\) is \(\boldsymbol{16}\).