Question

cho \(x=\sqrt[3]{\sqrt{2}-1}-\sqrt[3]{\sqrt{2}+1}\)

tính: \(A=x^3+3x+5\)

Answer 1

Đặt a=\(\sqrt[3]{\sqrt{2}-1}\Rightarrow x^3=\sqrt{2}-1\)

b=\(\sqrt[3]{\sqrt{2}+1}\Rightarrow b^3=\sqrt{2}+1\)

\(\Rightarrow x=a-b\)

\(\Rightarrow x^3=\left(a-b\right)^3\)

\(\Leftrightarrow x^3=a^3-3a^2b+3ab^3-b^3\)

\(\Leftrightarrow x^3=a^3+b^3-3ab\left(a-b\right)\)

\(\Leftrightarrow x^3=\sqrt{2}-1-\sqrt{2}-1-3\sqrt[3]{\sqrt{2}-1}\cdot\sqrt[3]{\sqrt{2}+1}x\)

\(\Leftrightarrow x^3=-2-3\sqrt[3]{2-1}x\)

\(\Leftrightarrow x^3=-2-3x\)

\(\Leftrightarrow x^3+3x=-2\)

\(\Leftrightarrow x^3+3x+5=3\)

Hay A=3

Answer 2

\(x^3=\sqrt{2}-1-\sqrt{2}-1-3x\sqrt[3]{2-1} \)

<=>\(x^3+3x+2=0\)

<=>\(x^3+3x+5=3\)

<=>A=3