4) \( \frac{b^{-3}}{4 b a^{4} \cdot 3 a^{4} b^{2}} \)

To simplify the expression \( \frac{b^{-3}}{4 b a^{4} \cdot 3 a^{4} b^{2}} \), we first simplify the denominator: 1. Combine the constants: \( 4 \cdot 3 = 12 \). 2. Combine like terms for \( b \): \( b \cdot b^{2} = b^{3} \). 3. The \( a \) terms multiply: \( a^{4} \cdot a^{4} = a^{8} \). Now the denominator simplifies to \( 12 b^{3} a^{8} \). We can rewrite our expression as: \[ \frac{b^{-3}}{12 b^{3} a^{8}} \] Next, using the properties of exponents, we can simplify the expression: \[ b^{-3} = \frac{1}{b^{3}} \] Thus, we can write: \[ \frac{1}{b^{3}} \cdot \frac{1}{12 b^{3} a^{8}} = \frac{1}{12 b^{3} b^{3} a^{8}} = \frac{1}{12 b^{6} a^{8}} \] Finally, we conclude that: \[ \frac{b^{-3}}{4 b a^{4} \cdot 3 a^{4} b^{2}} = \frac{1}{12 b^{6} a^{8}} \]