Lời giải:

Đặt \(\sqrt[3]{1+\frac{\sqrt{84}}{9}}=a; \sqrt[3]{1-\frac{\sqrt{84}}{9}}=b\)

Khi đó:

\(a^3+b^3=1+\frac{\sqrt{84}}{9}+1-\frac{\sqrt{84}}{9}=2\)

\(ab=\sqrt[3]{\left(1+\frac{\sqrt{84}}{9}\right)\left(1-\frac{\sqrt{84}}{9}\right)}=\sqrt[3]{1-\frac{84}{81}}=\frac{-1}{3}\)

Suy ra:

\(D^3=(a+b)^3=a^3+b^3+3ab(a+b)=2+3.\frac{-1}{3}.D\)

\(\Leftrightarrow D^3=2-D\Leftrightarrow D^3+D-2=0\)

\(\Leftrightarrow D^2(D-1)+D(D-1)+2(D-1)=0\)

\(\Leftrightarrow (D-1)(D^2+D+2)=0\)

Dễ thấy \(D^2+D+2>0\Rightarrow D-1=0\Leftrightarrow D=1\)

Vậy $D$ là một số nguyên.

Đặt \(A=\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

\(\Rightarrow A^3=1+\dfrac{\sqrt{84}}{9}+1-\dfrac{\sqrt{84}}{9}+3A.\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}.\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

\(\Leftrightarrow A^3=2+3A.\sqrt[3]{1-\dfrac{84}{81}}\)

\(\Leftrightarrow A^3=2+3A.\sqrt[3]{-\dfrac{3}{81}}=2+3A.\sqrt[3]{-\dfrac{1}{27}}\)

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

\(\Leftrightarrow A^3+A-2=0\)

\(\Leftrightarrow\left(A-1\right)\left(A^2+A+2\right)=0\)

Dể thấy \(A^2+A+2=\left(A+\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\)

\(\Rightarrow A-1=0\Leftrightarrow A=1\)

Vậy \(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\) là số nguyên (đpcm)

Bấm thử máy tính nó có ra số nguyên đâu em? Xem lại đề bài nhé.

Đặt \(\sqrt[3]{1+\frac{\sqrt{84}}{9}}=a;\sqrt[3]{1-\frac{\sqrt{84}}{9}}=b\Rightarrow x=a+b;a^3+b^3=2;ab=-\frac{1}{3}\)

Ta có:\(x^3=\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)

\(\Rightarrow x^3=2-x\Leftrightarrow x^3+x-2=0\Leftrightarrow\left(x-1\right).\left(x^2+x+2\right)=0\)

\(\Leftrightarrow x=1\).Vì \(x^2+x+2=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}>0\)

Từ đó suy ra điều phải chứng minh

~~~~~~~~~~~ Chúc bạn hok tốt~~~~~~~~~~~~

Tính giá trị của biểu thức \(P=x^3+y^3-3\left(x+y\right)+2004\)

biết \(x=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)và \(y=\sqrt[3]{17+12\sqrt{2}}+\sqrt[3]{17-12\sqrt{2}}\)

Đặt \(\sqrt[3]{1+\frac{\sqrt{84}}{9}}=a\);\(\sqrt[3]{1-\frac{\sqrt{84}}{9}}=b\)

\(\Rightarrow x=a+b;a^3+b^3=2;ab=-\frac{1}{3}\)

Ta có: \(x^3=\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)

\(\Rightarrow x^3=2-x\Leftrightarrow x^3+x-2=0\Leftrightarrow\left(x-1\right)\left(x^2+x+2\right)=0\)

\(\Leftrightarrow x=1\).vì \(x^2+x+2=0=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}>0\)

=> đpcm

P/s tham khảo

Đặt \(A=\sqrt[3]{1+\frac{\sqrt{84}}{9}}+\sqrt[3]{1-\frac{\sqrt{84}}{9}}\)

\(\Rightarrow A^3=1+\frac{\sqrt{84}}{9}+1-\frac{\sqrt{84}}{9}+3.\sqrt[3]{\left(1+\frac{\sqrt{84}}{9}\right)^2\left(1-\frac{\sqrt{84}}{9}\right)}+3.\sqrt[3]{\left(1+\frac{\sqrt{84}}{9}\right)\left(1-\frac{\sqrt{84}}{9}\right)^2}\)

\(A^3=2+3.\sqrt[3]{-\frac{1}{27}.\left(1+\frac{\sqrt{84}}{9}\right)}+3.\sqrt[3]{-\frac{1}{27}.\left(1-\frac{\sqrt{84}}{9}\right)}\)

      \(=2-\left(\sqrt[3]{\left(1+\frac{\sqrt{84}}{9}\right)}+\sqrt[.3]{\left(1-\frac{\sqrt{84}}{9}\right)}\right)\)

 \(A^3=2-A\Leftrightarrow\left(A-1\right)\left(A^2+A+2\right)=0\Rightarrow A=1\)

Đặt \(A=\sqrt[3]{\frac{9+2\sqrt{21}}{9}}+\sqrt[3]{\frac{9-2\sqrt{21}}{9}}\)

\(A^3=\frac{9+2\sqrt{21}+9-2\sqrt{21}}{9}+3\sqrt[3]{\frac{9^2-4\cdot21}{9^2}}A\)

\(A^3-2+A=0\Leftrightarrow\left(A-1\right)\left(A^2+A+1\right)+A-1=0\Leftrightarrow\left(A-1\right)\left(A^2+A+2\right)=0\)

\(\Rightarrow A=1\)(ĐPCM)

Tính A= \(\frac{\sqrt{2}\left(3+\sqrt{5}\right)}{2\sqrt{2}+\sqrt{3+\sqrt{5}}}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)}{2\sqrt{2}-\sqrt{3-\sqrt{ }}5}\)