If $16 x^2+y^2=48$ and $x y=2, x, y > 0$, then the value of $\left(64 x^3+y^3\right)$ is:

320

(a + b)³ = a³ + b³ + 3(ab)(a + b)

= 16x² + y² = 48

⇒ (4x)² + y² = 48

Add 8xy on both sides,

= (4x)² + y² + 8xy = 48 + 8xy

= (4x+y)² = 48 + 16

= (4x+y)² = 64

= 4x + y = 8

(a + b)³ = a³ + b³ + 3(ab)(a + b)

= (4x+y)³ = 8³

= 64x³ + y³ + 3(4xy)(4x+y) = 512

= 64x³ + y³ + (3 × 8 × 8) = 512

= 64x³ + y³ = 512 – 192 = 320