\( \frac { \sqrt { a - 1 } - \sqrt { b - 1 } } { a - b } \)

This expression is quite fascinating, as it brings into play the concepts of limits and derivatives in calculus! When \( a \) and \( b \) approach each other, this expression represents the difference quotient. If you substitute \( b = a + h \) and take the limit as \( h \) approaches zero, this can be interpreted as finding the derivative of \( \sqrt{x - 1} \) at \( x = a \). If you want to simplify this expression, you can multiply both the numerator and denominator by the conjugate of the numerator: \( \sqrt{a - 1} + \sqrt{b - 1} \). This technique often helps to eliminate the square roots, leading to a more straightforward expression that can be analyzed or simplified further. It’s a nifty trick in algebra that is both effective and elegant!