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1. Expand each term:

   • $3b^3(-2a)^2 = 3b^3 \cdot 4a^2 = 12a^2b^3$
   • $\frac{1}{2}a^2b(-2b)^2 = \frac{1}{2}a^2b \cdot 4b^2 = 2a^2b^3$
   • $-6(-ab)^2\left(-\frac{1}{2}b\right) = -6(a^2b^2)\left(-\frac{1}{2}b\right) = 3a^2b^3$

2. Combine all terms: $12a^2b^3 + 2a^2b^3 + 3a^2b^3 = 17a^2b^3$

Final Answer: $17a^2b^3$

Moving to Question 2:

$3m(2n-n)^2 - 4mn^2 + (-3m)(m-n)^2 - mn^2$

1. Simplify each term:

   • $2n-n = n$, so $(2n-n)^2 = n^2$
   • $3m(2n-n)^2 = 3m \cdot n^2 = 3mn^2$
   • $-4mn^2$ stays as it is.
   • $(-3m)(m-n)^2 = (-3m)(m^2 - 2mn + n^2) = -3m^3 + 6m^2n - 3mn^2$
   • $-mn^2$ stays as it is.

2. Combine all terms:

   • $3mn^2 - 4mn^2 - 3mn^2 - mn^2 = -5mn^2$
   • The cubic term is $-3m^3$, and the quadratic term is $6m^2n$.

Final Answer: $-3m^3 + 6m^2n - 5mn^2$

For Question 3:

$2x\left(-\frac{1}{2}y\right)^3 + \frac{1}{27}(m^2 + 2m^2)^3 - \frac{1}{4}x(-3y^3) - m^6$

1. Simplify each term:

   • $\left(-\frac{1}{2}y\right)^3 = -\frac{1}{8}y^3$, so $2x\left(-\frac{1}{2}y\right)^3 = 2x \cdot \left(-\frac{1}{8}y^3\right) = -\frac{1}{4}xy^3$
   • $m^2 + 2m^2 = 3m^2$, so $\frac{1}{27}(m^2 + 2m^2)^3 = \frac{1}{27}(3m^2)^3 = \frac{1}{27} \cdot 27m^6 = m^6$
   • $-\frac{1}{4}x(-3y^3) = -\frac{1}{4}x \cdot (-3y^3) = \frac{3}{4}xy^3$
   • $-m^6$ stays as it is.

2. Combine all terms:

   • The $xy^3$-terms: $-\frac{1}{4}xy^3 + \frac{3}{4}xy^3 = \frac{1}{2}xy^3$
   • The $m^6$-terms: $m^6 - m^6 = 0$

Final Answer: $\frac{1}{2}xy^3$

Would you like to proceed with more of these problems or review any specific part of the solution?

5 Related Questions:

1. How can you simplify expressions involving negative exponents?
2. What is the general rule for combining like terms in monomials?
3. How do you expand and simplify expressions with parentheses raised to a power?
4. Why is it important to calculate each term step by step in long expressions?
5. How do coefficients affect the simplification of polynomial terms?

Tip:

Always carefully distribute powers and negative signs when dealing with parentheses to avoid errors.