Let x, y > 0 . If x ^ 3 y ^ 2 = 2 ^ 15 , then the least value of 3x...

Using $A . M \ge G . M$, we get

$\dfrac{x + x + x + y + y}{5} \ge {x}^{3} \cdot {\left(y\right)}^{2}5$

$⇒ \dfrac{3x + 2y}{5} \ge {x}^{3}{y}^{2}5$

$⇒ \dfrac{3x + 2y}{5} \ge \left({2}^{15}\right)5 \left(∵ {x}^{3}{y}^{2} = {2}^{15}\right)$

$⇒ 3x + 2y \ge 5 \cdot {\left(2\right)}^{15 / 5}$

$⇒ 3x + 2y ⩾ 5 \times {2}^{3}$

$⇒ 3x + 2y ⩾ 40$

$∴$ The minimum value of $3x + 2y$ is $40$