4) \( \left(\frac{3 a^{6} b^{5}}{6 a^{7} b^{4}}\right)^{4} \)

To simplify the expression \( \left(\frac{3 a^{6} b^{5}}{6 a^{7} b^{4}}\right)^{4} \), we start with the fraction inside the parentheses: 1. Simplify the constants: \( \frac{3}{6} = \frac{1}{2} \). 2. For the variables, we apply the laws of exponents: - For \( a \): \( \frac{a^{6}}{a^{7}} = a^{6-7} = a^{-1} \). - For \( b \): \( \frac{b^{5}}{b^{4}} = b^{5-4} = b^{1} = b \). Now we can rewrite the fraction: \[ \frac{3 a^{6} b^{5}}{6 a^{7} b^{4}} = \frac{1}{2} a^{-1} b \] Next, we raise this to the power of 4: \[ \left(\frac{1}{2} a^{-1} b\right)^{4} = \left(\frac{1}{2}\right)^{4} (a^{-1})^{4} b^{4} \] Calculating each part: - \( \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \) - \( (a^{-1})^{4} = a^{-4} \) - \( b^{4} = b^{4} \) Now we combine these results: \[ \frac{1}{16} a^{-4} b^{4} = \frac{b^{4}}{16 a^{4}} \] So the final simplified expression is: \[ \frac{b^{4}}{16 a^{4}} \]