ii) \( \frac{6}{\sqrt{5}-1}= \)

To simplify \( \frac{6}{\sqrt{5}-1} \), we multiply the numerator and the denominator by the conjugate of the denominator, \( \sqrt{5}+1 \). This gives us: \[ \frac{6(\sqrt{5}+1)}{(\sqrt{5}-1)(\sqrt{5}+1)} = \frac{6(\sqrt{5}+1)}{5-1} = \frac{6(\sqrt{5}+1)}{4} = \frac{3(\sqrt{5}+1)}{2} \] So, \( \frac{6}{\sqrt{5}-1} = \frac{3(\sqrt{5}+1)}{2} \). For those curious about mathematical procedures, this method of rationalizing the denominator is widely used in algebra. By eliminating the square root from the denominator, we put the expression in a more manageable form, making it easier to work with in further calculations or applications. In practical applications, this technique is not just a neat trick; it enables clearer communication of mathematical ideas, especially in fields like engineering and physics where precision is crucial. Simplified expressions help avoid misunderstandings and calculations which might compound errors later on.