c) \( \left(\frac{1}{6}\right)^{2}+\left(\frac{6}{4}\right)^{-2} \)

To solve the expression \( \left(\frac{1}{6}\right)^{2}+\left(\frac{6}{4}\right)^{-2} \), we start with the first term. Squaring \( \frac{1}{6} \) gives us \( \frac{1}{36} \). Next, for the second term, we recognize that \( \left(\frac{6}{4}\right)^{-2} \) is the same as \( \left(\frac{3}{2}\right)^{-2} \). The negative exponent indicates we should take the reciprocal and then square it, resulting in \( \left(\frac{2}{3}\right)^{2} = \frac{4}{9} \). Now, we need a common denominator to add \( \frac{1}{36} \) and \( \frac{4}{9} \). The least common multiple of 36 and 9 is 36. To convert \( \frac{4}{9} \) to have the same denominator, we multiply by \( \frac{4}{4} \) giving us \( \frac{16}{36} \). Finally, we can combine the two fractions: \[ \frac{1}{36} + \frac{16}{36} = \frac{17}{36} \] Thus, the final result is \[ \left(\frac{1}{6}\right)^{2}+\left(\frac{6}{4}\right)^{-2} = \frac{17}{36}. \]