1. **State the problem:** Calculate $\log_6 \left( \frac{216 \cdot 1296 \cdot \sqrt[3]{6}}{\sqrt{1296}} \right)$. 2. **Recall logarithm properties:** - $\log_b(xy) = \log_b x + \log_b y$ - $\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y$ - $\log_b (x^r) = r \log_b x$ 3. **Simplify the expression inside the logarithm:** - $216 = 6^3$ because $6^3 = 216$ - $1296 = 6^4$ because $6^4 = 1296$ - $\sqrt[3]{6} = 6^{1/3}$ - $\sqrt{1296} = \sqrt{6^4} = 6^{4/2} = 6^2$ 4. **Rewrite the argument using exponents:** $$\frac{216 \cdot 1296 \cdot \sqrt[3]{6}}{\sqrt{1296}} = \frac{6^3 \cdot 6^4 \cdot 6^{1/3}}{6^2}$$ 5. **Combine the powers of 6 in numerator:** $$6^3 \cdot 6^4 \cdot 6^{1/3} = 6^{3+4+\frac{1}{3}} = 6^{7 + \frac{1}{3}} = 6^{\frac{21}{3} + \frac{1}{3}} = 6^{\frac{22}{3}}$$ 6. **Divide by denominator:** $$\frac{6^{\frac{22}{3}}}{6^2} = 6^{\frac{22}{3} - 2} = 6^{\frac{22}{3} - \frac{6}{3}} = 6^{\frac{16}{3}}$$ 7. **Apply logarithm:** $$\log_6 \left(6^{\frac{16}{3}}\right) = \frac{16}{3} \log_6 6 = \frac{16}{3} \times 1 = \frac{16}{3}$$ **Final answer:** $$\boxed{\frac{16}{3}}$$