Question

if x = root 7 minus root 5 , y = root 5 minus root 3, Z equals to root 3 minus root 7, then find the value of x cube + y cube + Z cube minus 2 x y z

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{x=\sqrt{7}-\sqrt{5}}$

$\mathsf{y=\sqrt{5}-\sqrt{3}}$

$\mathsf{z=\sqrt{3}-\sqrt{7}}$

$\underline{\textsf{To find:}}$

$\textsf{The value of}$

$\mathsf{x^3+y^3+z^3-2\,xyz}$

$\underline{\textsf{Solution:}}$

$\underline{\mathsf{Concept\;used:}}$

$\boxed{\mathsf{If\;a+b+c=0\;then\;a^3+b^3+c^3=3\,abc}}$

$\mathsf{Consider,}$

$\mathsf{x+y+z=\sqrt{7}-\sqrt{5}+\sqrt{5}-\sqrt{3}+\sqrt{3}-\sqrt{7}=0}$

$\mathsf{Then,}$

$\mathsf{x^3+y^3+z^3-3\,xyz=0}$

$\mathsf{x^3+y^3+z^3-2\,xyz-xyz=0}$

$\mathsf{x^3+y^3+z^3-2\,xyz=xyz}$

$\mathsf{x^3+y^3+z^3-2\,xyz=(\sqrt{7}-\sqrt{5})(\sqrt{5}-\sqrt{3})(\sqrt{3}-\sqrt{7})}$

$\mathsf{x^3+y^3+z^3-2\,xyz=(\sqrt{7}\sqrt{5}-\sqrt{7}\sqrt{3}-5+\sqrt{5}\sqrt{3})(\sqrt{3}-\sqrt{7})}$

$\mathsf{x^3+y^3+z^3-2\,xyz=\sqrt{7}\sqrt{5}\sqrt{3}-3\sqrt{7}-5\sqrt{3}+3\sqrt{5}-7\sqrt{5}+7\sqrt{3}+5\sqrt{7}-\sqrt{7}\sqrt{5}\sqrt{3}}$

$\mathsf{x^3+y^3+z^3-2\,xyz=-3\sqrt{7}-5\sqrt{3}+3\sqrt{5}-7\sqrt{5}+7\sqrt{3}+5\sqrt{7}}$

$\mathsf{x^3+y^3+z^3-2\,xyz=-4\sqrt{5}+2\sqrt{3}+2\sqrt{7}}$

$\mathsf{x^3+y^3+z^3-2\,xyz=2\sqrt{3}+2\sqrt{7}-4\sqrt{5}}$

$\implies\boxed{\mathsf{x^3+y^3+z^3-2\,xyz=2(\sqrt{3}+\sqrt{7}-2\sqrt{5})}}$

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Answer 2

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