1. **State the problem:** Simplify the expression $$(9abx^4 - cx^7)(9abx^4 + cx^7)$$. 2. **Formula used:** This is a product of conjugates, which follows the difference of squares formula: $$ (A - B)(A + B) = A^2 - B^2 $$ where $A = 9abx^4$ and $B = cx^7$. 3. **Apply the formula:** $$ (9abx^4)^2 - (cx^7)^2 $$ 4. **Calculate each square:** $$ (9abx^4)^2 = 9^2 \cdot a^2 \cdot b^2 \cdot (x^4)^2 = 81a^2b^2x^8 $$ $$ (cx^7)^2 = c^2 \cdot (x^7)^2 = c^2x^{14} $$ 5. **Write the simplified expression:** $$ 81a^2b^2x^8 - c^2x^{14} $$ 6. **Final answer:** $$ \boxed{81a^2b^2x^8 - c^2x^{14}} $$