1. **State the problem:** Simplify the expression $$6 \left( \frac{x^3 y^{12}}{z^4} \right)^2 \cdot \frac{x^5}{1}$$. 2. **Recall the exponent rules:** - When raising a power to another power, multiply the exponents: $\left(a^m\right)^n = a^{mn}$. - When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$. - When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. 3. **Apply the power to each factor inside the parentheses:** $$\left( \frac{x^3 y^{12}}{z^4} \right)^2 = \frac{(x^3)^2 (y^{12})^2}{(z^4)^2} = \frac{x^{3 \cdot 2} y^{12 \cdot 2}}{z^{4 \cdot 2}} = \frac{x^6 y^{24}}{z^8}$$ 4. **Rewrite the entire expression:** $$6 \cdot \frac{x^6 y^{24}}{z^8} \cdot \frac{x^5}{1} = 6 \cdot \frac{x^6 y^{24} x^5}{z^8}$$ 5. **Combine the $x$ terms in the numerator:** $$x^6 \cdot x^5 = x^{6+5} = x^{11}$$ 6. **Final simplified expression:** $$\frac{6 x^{11} y^{24}}{z^8}$$ **Answer:** $$\boxed{\frac{6 x^{11} y^{24}}{z^8}}$$