1. **State the problem:** Simplify the expression $$\left[(c^2 d)^4 \times (c^3 d^5)^2\right]^3 \div (c^4 d^7)^5$$. 2. **Recall exponent rules:** - Power of a power: $$(a^m)^n = a^{mn}$$ - Product of powers: $$a^m \times a^n = a^{m+n}$$ - Quotient of powers: $$\frac{a^m}{a^n} = a^{m-n}$$ - Power of a product: $$(ab)^n = a^n b^n$$ 3. **Simplify inside the brackets:** - First term: $$(c^2 d)^4 = c^{2 \times 4} d^{1 \times 4} = c^8 d^4$$ - Second term: $$(c^3 d^5)^2 = c^{3 \times 2} d^{5 \times 2} = c^6 d^{10}$$ 4. **Multiply the two terms inside the brackets:** $$c^8 d^4 \times c^6 d^{10} = c^{8+6} d^{4+10} = c^{14} d^{14}$$ 5. **Raise the product to the power 3:** $$\left(c^{14} d^{14}\right)^3 = c^{14 \times 3} d^{14 \times 3} = c^{42} d^{42}$$ 6. **Simplify the denominator:** $$(c^4 d^7)^5 = c^{4 \times 5} d^{7 \times 5} = c^{20} d^{35}$$ 7. **Divide numerator by denominator:** $$\frac{c^{42} d^{42}}{c^{20} d^{35}} = c^{42-20} d^{42-35} = c^{22} d^{7}$$ **Final answer:** $$c^{22} d^{7}$$ which corresponds to option $\boxed{c^{22} d^{7}}$.