Đặt \(x=\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt[]{3}}\)

\(\Rightarrow x^3=52+3\sqrt[3]{\left(26+15\sqrt[]{3}\right)\left(26-15\sqrt[]{3}\right)}.x\)

\(\Leftrightarrow x^3=52+3x\)

\(\Leftrightarrow x^3-3x-52=0\)

\(\Leftrightarrow\left(x-4\right)\left(x^2+4x+13\right)=0\)

\(\Leftrightarrow\left(x-4\right)\left[\left(x+2\right)^2+9\right]=0\)

\(\Leftrightarrow x=4\)

\(=\sqrt{\left(2-\sqrt{3}\right)^2\left(26+15\sqrt{3}\right)}-\sqrt{\left(2+\sqrt{3}\right)^2\left(26-15\sqrt{3}\right)}=\)

\(=\sqrt{\left(7-4\sqrt{3}\right)\left(26+15\sqrt{3}\right)}-\sqrt{\left(7+4\sqrt{3}\right)\left(26-15\sqrt{3}\right)=}\)

\(=\sqrt{7.26+7.15\sqrt{3}-4.26\sqrt{3}-180}-\sqrt{7.26-7.15\sqrt{3}+4.26\sqrt{3}-180}=\)

\(=\sqrt{4+\sqrt{3}}-\sqrt{4-\sqrt{3}}\)

Lời giải:

Gọi biểu thức trên là $A$

Đặt \(\sqrt[3]{15\sqrt{3}-26}=a; \sqrt[3]{15\sqrt{3}+26}=b\). Ta có:

\(a^3-b^3=-52\)

\(ab=-1\)

\(A^3=(a-b)^3=a^3-3ab(a-b)-b^3=-52+3A\)

\(\Leftrightarrow A^3-3A+52=0\)

\(\Leftrightarrow A^2(A+4)-4A(A+4)+13(A+4)=0\)

\(\Leftrightarrow (A+4)(A^2-4A+13)=0\)

Dễ thấy $A^2-4A+13>0$ nên $A+4=0$

$\Leftrightarrow A=-4$

\(a=\sqrt[3]{15\sqrt{3}+26}+\sqrt[3]{15\sqrt{3}-26}\)

\(a^3=30\sqrt{3}+3a.\sqrt[3]{15^2.3-26^2}=30\sqrt{3}-3a\)

\(\Leftrightarrow a^3+3a-30\sqrt{3}=0\)

\(\Leftrightarrow\left(a-2\sqrt{3}\right)\left(a^2+2\sqrt{3}a+15\right)=0\)

\(\Rightarrow a=2\sqrt{3}\)

\(\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}}\)

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

=\(\sqrt{3}+2+\sqrt{3}-2=2\sqrt{3}\)

\(A=\sqrt[3]{26+15\sqrt{3}}-\sqrt[3]{26-15\sqrt{3}}=\sqrt[3]{8+3.4.\sqrt{3}+3.2.3+3\sqrt{3}}-\sqrt[3]{8-3.4.\sqrt{3}+3.2.3-3\sqrt{3}}=\sqrt[3]{\left(2+\sqrt{3}\right)^3}-\sqrt[3]{\left(2-\sqrt{3}\right)^3}=2+\sqrt{3}-2+\sqrt{3}=2\sqrt{3}\)

\(\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}}=\sqrt[3]{8+12\sqrt{3}+18+3\sqrt{3}}+\sqrt[3]{8-12\sqrt{3}+18-3\sqrt{3}}=\sqrt[3]{\left(2+\sqrt{3}\right)^3}+\sqrt[3]{\left(2-\sqrt{3}\right)^3}=2+\sqrt{3}+2-\sqrt{3}=4\)

\(=\sqrt[3]{2^3+3.2^2.\sqrt{3}+3.2.\left(\sqrt{3}\right)^2+\left(\sqrt{3}\right)^3}+\sqrt[3]{2^3-3.2^2.\sqrt{3}+3.2.\left(\sqrt{3}\right)^2+\left(-\sqrt{3}\right)^3}\)

\(=\sqrt[3]{\left(2+\sqrt{3}\right)^3}+\sqrt[3]{\left(2-\sqrt{3}\right)^3}=2+\sqrt{3}+2-\sqrt{3}=4\)