Find the Value of \(64x^3-125z^3\), if \(4x-5z=16\) and \(xz=12\)
Solution:
Using identity:
\[ a^3-b^3=(a-b)^3+3ab(a-b) \]
Here,
\[ a=4x,\quad b=5z \]
\[ a-b=16 \]
\[ ab=(4x)(5z)=20xz=20(12)=240 \]
\[ 64x^3-125z^3 = (16)^3+3(240)(16) \]
\[ = 4096+11520 \]
\[ =15616 \]