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Answer :
We start with the expression
$$
(3x^2y) \cdot (2xy^3) \cdot (4x^2y^2).
$$
**Step 1: Multiply the coefficients**
Multiply the numerical coefficients together:
$$
3 \times 2 \times 4 = 24.
$$
**Step 2: Multiply the \( x \) terms**
When multiplying like bases, add the exponents. For the \( x \) terms, the exponents are 2, 1, and 2:
$$
x^2 \cdot x^1 \cdot x^2 = x^{2+1+2} = x^5.
$$
**Step 3: Multiply the \( y \) terms**
Similarly, for the \( y \) terms, the exponents are 1, 3, and 2:
$$
y^1 \cdot y^3 \cdot y^2 = y^{1+3+2} = y^6.
$$
**Step 4: Combine all parts**
Put together the results from Steps 1–3:
$$
(3x^2y) \cdot (2xy^3) \cdot (4x^2y^2) = 24x^5y^6.
$$
Thus, the simplified form of the given expression is
$$
\boxed{24x^5y^6}.
$$
This corresponds to option D.
$$
(3x^2y) \cdot (2xy^3) \cdot (4x^2y^2).
$$
**Step 1: Multiply the coefficients**
Multiply the numerical coefficients together:
$$
3 \times 2 \times 4 = 24.
$$
**Step 2: Multiply the \( x \) terms**
When multiplying like bases, add the exponents. For the \( x \) terms, the exponents are 2, 1, and 2:
$$
x^2 \cdot x^1 \cdot x^2 = x^{2+1+2} = x^5.
$$
**Step 3: Multiply the \( y \) terms**
Similarly, for the \( y \) terms, the exponents are 1, 3, and 2:
$$
y^1 \cdot y^3 \cdot y^2 = y^{1+3+2} = y^6.
$$
**Step 4: Combine all parts**
Put together the results from Steps 1–3:
$$
(3x^2y) \cdot (2xy^3) \cdot (4x^2y^2) = 24x^5y^6.
$$
Thus, the simplified form of the given expression is
$$
\boxed{24x^5y^6}.
$$
This corresponds to option D.
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