Question

Tính gtrị biểu thức

A=(x-y)³+3(x-y)(xy+1),biết x=\(\sqrt[3]{2+\sqrt{3}}-\sqrt[3]{2-\sqrt{3}}\)

=\(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5-2}}\)

Answer 1

Lời giải:

Áp dụng HĐT $(a-b)^3=a^3-3a^2b+3ab^2-b^3=a^3-3ab(a-b)-b^3$

\(x^3=2+\sqrt{3}-3\sqrt[3]{(2+\sqrt{3})(2-\sqrt{3})}x-(2-\sqrt{3})\)

\(\Leftrightarrow x^3=2\sqrt{3}-3x\)

\(\Leftrightarrow x^3+3x=2\sqrt{3}\)

\(y^3=\sqrt{5}+2-3\sqrt[3]{(\sqrt{5}+2)(\sqrt{5}-2)}y-(\sqrt{5}-2)\)

\(\Leftrightarrow y^3=4-3y\Leftrightarrow y^3+3y=4\)

Do đó:
\(A=(x-y)^3+3(x-y)(xy+1)=x^3-3xy(x-y)-y^3+3[xy(x-y)+(x-y)]\)

\(=x^3-y^3+3(x-y)=(x^3+3x)-(y^3+3y)=2\sqrt{3}-4\)