Question:
Find the value of $64 x^{3}-125 z^{3}$, if $4 x-5 z=16$ and $x z=12$
Solution:
Given, $64 x^{3}-125 z^{3}$
Here, $4 x-5 z=16$ and $x z=12$
Cubing $4 x-5 z=16$ on both sides
$(4 x-5 z)^{3}=16^{3}$
We know that, $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$
$(4 x)^{3}-(5 z)^{3}-3(4 x)(5 z)(4 x-5 z)=16^{3}$
$64 x^{3}-125 z^{3}-60(x z)(16)=4096$
$64 x^{3}-125 z^{3}-60(12)(16)=4096$
$64 x^{3}-125 z^{3}-11520=4096$
$64 x^{3}-125 z^{3}=4096+11520$
$64 x^{3}-125 z^{3}=15616$
The value of $64 x^{3}-125 z^{3}=15616$
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