13. \( \left(6 a^{2} b\right)^{3} \cdot\left(\frac{1}{3} a b c\right)^{2} \)

To simplify \( \left(6 a^{2} b\right)^{3} \cdot\left(\frac{1}{3} a b c\right)^{2} \), we will apply the power of a product rule and the properties of exponents. First, simplify \( \left(6 a^{2} b\right)^{3} \): \[ (6^3)(a^{2 \cdot 3})(b^3) = 216 a^{6} b^{3} \] Next, simplify \( \left(\frac{1}{3} a b c\right)^{2} \): \[ \left(\frac{1}{3}\right)^{2}(a^{2})(b^{2})(c^{2}) = \frac{1}{9} a^{2} b^{2} c^{2} \] Now, multiply the two results: \[ (216 a^{6} b^{3}) \cdot \left(\frac{1}{9} a^{2} b^{2} c^{2}\right) \] This can be calculated as follows: \[ = 216 \cdot \frac{1}{9} (a^{6} \cdot a^{2})(b^{3} \cdot b^{2}) c^{2} \] \[ = 24 a^{6 + 2} b^{3 + 2} c^{2} \] \[ = 24 a^{8} b^{5} c^{2} \] Thus, the simplified expression is: \[ \boxed{24 a^{8} b^{5} c^{2}} \]