1. Let's solve the first problem from the user's message: Simplify the expression $(x - 7y) \cdot 3(3x - 7)y \cdot (3x - 7) \cdot y$. 2. The problem involves multiplication of algebraic expressions. We use the distributive property and multiplication of terms. 3. First, rewrite the expression clearly: $$ (x - 7y) \times 3(3x - 7)y \times (3x - 7) \times y $$ 4. Simplify step-by-step: - Multiply $3(3x - 7)y$ to get $3y(3x - 7) = 9xy - 21y$ 5. Now the expression is: $$ (x - 7y) \times (9xy - 21y) \times (3x - 7) \times y $$ 6. Next, multiply $(x - 7y)$ and $(9xy - 21y)$: $$ (x)(9xy) + (x)(-21y) + (-7y)(9xy) + (-7y)(-21y) $$ $$ = 9x^2 y - 21xy - 63 x y^2 + 147 y^2 $$ 7. So the product is: $$ 9x^2 y - 21xy - 63 x y^2 + 147 y^2 $$ 8. Now multiply this result by $(3x - 7)$: Multiply each term by $3x$ and then by $-7$: $$ (9x^2 y)(3x) + (-21xy)(3x) + (-63 x y^2)(3x) + (147 y^2)(3x) $$ $$ + (9x^2 y)(-7) + (-21xy)(-7) + (-63 x y^2)(-7) + (147 y^2)(-7) $$ 9. Calculate each term: $$ 27 x^3 y - 63 x^2 y - 189 x^2 y^2 + 441 x y^2 - 63 x^2 y + 147 x y + 441 x y^2 - 1029 y^2 $$ 10. Combine like terms: - $-63 x^2 y - 63 x^2 y = -126 x^2 y$ - $441 x y^2 + 441 x y^2 = 882 x y^2$ So the expression becomes: $$ 27 x^3 y - 126 x^2 y - 189 x^2 y^2 + 882 x y^2 + 147 x y - 1029 y^2 $$ 11. Finally, multiply the entire expression by $y$: $$ 27 x^3 y^2 - 126 x^2 y^2 - 189 x^2 y^3 + 882 x y^3 + 147 x y^2 - 1029 y^3 $$ 12. This is the fully simplified expression. **Final answer:** $$ 27 x^3 y^2 - 126 x^2 y^2 - 189 x^2 y^3 + 882 x y^3 + 147 x y^2 - 1029 y^3 $$