Math, asked by prakharmishra449, 1 year ago

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

Answers

Answered by MaheswariS
14

\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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Answered by mahendra15aug
2

Please mark my answer as the brainliest answer.

Hope it helps you.

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