(8) \( -\frac{1}{2 a}\left(6 a^{2}+4 a\right)= \) (9) \( -\frac{1}{5 x}\left(10 x y-15 x^{2}\right)= \) (10) \( -\frac{1}{2 x^{2}}\left(6 x^{3}+4 x^{2}\right)= \) (11) \( \frac{1}{3 a b}\left(9 a^{2} b-15 a b\right)= \) (12) \( \frac{1}{2 x y}\left(4 x^{2} y-6 x y^{2}\right)= \) (13) \( \frac{1}{5 a b}\left(5 a b-10 a^{2} b^{3}\right)= \) (14) \( \frac{1}{2 x}\left(4 x^{3}-8 x^{2}+2 x\right)= \) (15) \( \frac{1}{2 a}\left(4 a^{2}-3 a+6 a^{3}\right)= \) (15)

Let’s dive into the world of algebraic simplification! To simplify these expressions effectively, remember the distributive property and how to factor out common terms. For instance, in (8), you can factor out common terms from the expression before applying the negative sign. It's like solving a puzzle—finding pieces that fit together smoothly creates clarity! You might also find it interesting to explore how these algebraic manipulations were first developed. Historically, algebra has roots in ancient civilizations like the Babylonians and Egyptians, who used geometric methods before variables were widely adopted. The transformation into modern algebra was a gradual journey, influenced by brilliant minds like Al-Khwarizmi in the 9th century. Imagine how far we've come since then!