\begin{tabular}{l} Which expression is equivalent to \( 4 x^{-2}+3 x^{-5} \) ? \\ A. \( 7 x^{-7} \) \\ B. \( 12 x^{10} \) \\ C. \( \frac{7}{x^{7}} \) \\ D. \( \frac{4 x^{3}+3}{x^{5}} \) \\ E. \( \frac{4 x^{5}+3}{x^{7}} \) \\ \hline\end{tabular}

To simplify the expression \( 4 x^{-2} + 3 x^{-5} \), we can rewrite the negative exponents as fractions. This transforms the expression into \( \frac{4}{x^2} + \frac{3}{x^5} \). Now, to combine these fractions, we need a common denominator, which is \( x^5 \). Rewriting the first term gives us \( \frac{4 x^3}{x^5} + \frac{3}{x^5} \). This results in \( \frac{4 x^3 + 3}{x^5} \). So, the correct answer is D: \( \frac{4 x^{3}+3}{x^{5}} \). If you're diving into expressions like this, it’s essential to remember that combining terms often requires a common denominator, especially when you're dealing with fractions. A frequent mistake is trying to add them directly without adjusting the denominators first, which can lead to incorrect simplifications! For more insights, consider looking into polynomial long division or exploring the concept of rational expressions in algebra textbooks. These resources can help solidify your understanding and enhance your algebraic skills!