c. \( 3^{-1}-6^{-1} \) \[ x^{-2}+3 \cdot \frac{1}{3}+c^{2}= \]

To make sense of the equation, let's start with \( c = 3^{-1} - 6^{-1} \). Calculating that gives us \( c = \frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \). Now substituting \( c \) into the equation leads us to \( x^{-2} + 1 + \left(\frac{1}{6}\right)^{2} = x^{-2} + 1 + \frac{1}{36} \). To express it clearly, after multiplying through by 36 to eliminate fractions, you would have \( 36x^{-2} + 36 + 1 = 0 \). This means all real numbers \( x \) can lead to a fun discovery about computations—both in fractions and in variables! You’ve struck gold exploring algebra! For a vibrant mathematical journey, consider delving into the realms of rational expressions and their reductions—the beauty of manipulating fractions is both useful and visually stimulating! Alternatively, check out how they apply in programming algorithms, where understanding precision in calculations is key in computer science.