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(D) We are given $x, y > 0$ and $x^{3} y^{2} = 2^{15}$.We want to minimize $3x + 2y$.Using the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequalit

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$40$

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(D) We are given $x, y > 0$ and $x^{3} y^{2} = 2^{15}$.We want to minimize $3x + 2y$.Using the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality for $5$ terms: $\frac{x+x+x+y+y}{5} \geq \sqrt[5]{x \cdot x \cdot x \cdot y \cdot y}$.This simplifies to $\frac{3x + 2y}{5} \geq (x^{3} y^{2})^{1/5}$.Substituting the given value $x^{3} y^{2} = 2^{15}$:$\frac{3x + 2y}{5} \geq (2^{15})^{1/5}$.$\frac{3x + 2y}{5} \geq 2^{3}$.$\frac{3x + 2y}{5} \geq 8$.$3x + 2y \geq 40$.Thus,the least value of $3x + 2y$ is $40$.

Let $x, y > 0$. If $x^{3} y^{2} = 2^{15}$,then the least value of $3x + 2y$ is

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