Find the value of $64 x^{3} - 125 z^{3}, i f 4 x - 5 z = 16 a n d x z = 12.$

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Solution

4x - 5z = 16, xz = 12

Cubing both sides,

$(4 x - 5 z)^{3} = (16)^{3}$

$\Rightarrow (4 x)^{3} - (5 y)^{3} - 3 \times 4 x \times 5 z (4 x - 5 z) = 4096 \Rightarrow 64 x^{3} - 125 z^{3} - 3 \times 4 \times 5 \times x z (4 x - 5 z) = 4096 \Rightarrow 64 x^{3} - 125 z^{3} - 60 \times 12 \times 16 = 4096 \Rightarrow 64 x^{3} - 125 z^{3} - 11520 = 4096 \Rightarrow 64 x^{3} - 125 z^{3} = 4096 + 11520 = 15616$

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If $4 x - 5 y = 16$ and $x y = 12$ then the value of $64 x^{3} - 125 y^{3} =$ is _______

A. $4648$
B. $15616$
C. $4672$
D. $4696$
E. None of these

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