1. **State the problem:** Simplify the expression $$\frac{(-6a^{2}b)^{2}}{(3ab)^{2}} - (2a)^{3} : (-a) + 6 \left[-(-a^{2}b)^{2}\right]^{3} : (a^{5}b^{3})^{2}$$. 2. **Recall the rules:** - Power of a product: $$(xy)^n = x^n y^n$$ - Power of a power: $$(x^m)^n = x^{mn}$$ - Division of powers with same base: $$\frac{x^m}{x^n} = x^{m-n}$$ - Negative signs inside powers: $$(-x)^2 = x^2$$ - Order of operations: parentheses, exponents, multiplication/division, addition/subtraction. 3. **Calculate each part:** - First term numerator: $$(-6a^{2}b)^2 = (-6)^2 (a^{2})^2 b^2 = 36 a^{4} b^{2}$$ - First term denominator: $$(3ab)^2 = 3^2 a^2 b^2 = 9 a^{2} b^{2}$$ - First term fraction: $$\frac{36 a^{4} b^{2}}{9 a^{2} b^{2}} = \frac{\cancel{36}^4 \cancel{a^{4}}^{2} \cancel{b^{2}}^{2}}{\cancel{9}^1 \cancel{a^{2}}^{2} \cancel{b^{2}}^{2}} = 4 a^{2}$$ - Second term: $$(2a)^3 = 2^3 a^3 = 8 a^{3}$$ - Third term: $$-a$$ - Fourth term inside brackets: $$-(-a^{2}b)^2 = -((-1)^2 a^{4} b^{2}) = - (1 a^{4} b^{2}) = - a^{4} b^{2}$$ - Fourth term bracket cubed: $$\left(- a^{4} b^{2}\right)^3 = (-1)^3 a^{12} b^{6} = - a^{12} b^{6}$$ - Multiply by 6: $$6 \times (- a^{12} b^{6}) = -6 a^{12} b^{6}$$ - Fifth term denominator: $$(a^{5} b^{3})^{2} = a^{10} b^{6}$$ - Fourth term division: $$\frac{-6 a^{12} b^{6}}{a^{10} b^{6}} = -6 \frac{a^{12}}{a^{10}} \frac{b^{6}}{b^{6}} = -6 a^{2}$$ 4. **Combine all terms:** $$4 a^{2} - 8 a^{3} : (-a) - 6 a^{2}$$ 5. **Division in second term:** $$8 a^{3} : (-a) = \frac{8 a^{3}}{-a} = -8 a^{2}$$ 6. **Final expression:** $$4 a^{2} - (-8 a^{2}) - 6 a^{2} = 4 a^{2} + 8 a^{2} - 6 a^{2} = (4 + 8 - 6) a^{2} = 6 a^{2}$$ **Answer:** $$6 a^{2}$$