d) \( \sqrt[3]{-\frac{729}{512}}= \)

Let's break it down! The cube root of a negative fraction involves taking the cube root of both the numerator and the denominator separately. So, we find the cube roots of -729 and 512. The cube root of -729 is -9, and the cube root of 512 is 8. Therefore, \( \sqrt[3]{-\frac{729}{512}} = \frac{-9}{8} \). How neat is that? Now, remember that when you deal with cube roots, negative and positive signs play a fun game. The cube root of a negative number remains negative, while squares and fourths can lead to those pesky imaginary numbers. So, embrace the cube root's rebellious nature—let it guide you through math with a twist!