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

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

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

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

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) Ta có: \(M=\dfrac{2}{\sqrt{7}-\sqrt{6}}-\sqrt{28}+\sqrt{54}\)

\(=\dfrac{2\left(\sqrt{7}+\sqrt{6}\right)}{\left(\sqrt{7}-\sqrt{6}\right)\left(\sqrt{7}+\sqrt{6}\right)}-2\sqrt{7}+3\sqrt{6}\)

\(=2\sqrt{7}+2\sqrt{6}-2\sqrt{7}+3\sqrt{6}\)

\(=5\sqrt{6}\)

b) Ta có: \(N=\left(2-\sqrt{3}\right)\left(\sqrt{26+15\sqrt{3}}\right)-\left(2+\sqrt{3}\right)\sqrt{26-15\sqrt{3}}\)

\(=\dfrac{\left(2-\sqrt{3}\right)\sqrt{52+30\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt{52-30\sqrt{3}}}{\sqrt{2}}\)

\(=\dfrac{\left(2-\sqrt{3}\right)\sqrt{27+2\cdot3\sqrt{3}\cdot5+25}-\left(2+\sqrt{3}\right)\sqrt{27-2\cdot3\sqrt{3}\cdot5+25}}{\sqrt{2}}\)

\(=\dfrac{\left(2-\sqrt{3}\right)\sqrt{\left(3\sqrt{3}+5\right)^2}-\left(2+\sqrt{3}\right)\sqrt{\left(3\sqrt{3}-5\right)^2}}{\sqrt{2}}\)

\(=\dfrac{\left(2-\sqrt{3}\right)\left(3\sqrt{3}+5\right)-\left(2+\sqrt{3}\right)\left(3\sqrt{3}-5\right)}{\sqrt{2}}\)

\(=\dfrac{6\sqrt{3}+10-9-5\sqrt{3}-\left(6\sqrt{3}-10+9-5\sqrt{3}\right)}{\sqrt{2}}\)

\(=\dfrac{\sqrt{3}+1-\sqrt{3}+1}{\sqrt{2}}\)

\(=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)

Xét: \(A=\sqrt{26+15\sqrt{3}}\)  dễ thấy A > 0

\(\Leftrightarrow A^2=52-2\sqrt{26^2-15^2.3}=50\Leftrightarrow A=\sqrt{50}\)

Vậy: \(A=2+\sqrt{3}.\sqrt{26+15\sqrt{3}}-2\sqrt{3}.\sqrt{26-15\sqrt{3}}\)

\(=2+\sqrt{3}.A=2+\sqrt{3}.\sqrt{50}=5\sqrt{6}+10\sqrt{2}\)

Ý tưởng; Dựa vào "thế hình" của đề bài, ta dự đoán biểu thức trong căn sẽ phân tích được thành lũy thừa bậc 3 của một biểu thức có dạng \(a+\sqrt{3}\left(a\inℤ\right)\). Ta thấy:

 \(\left(a+\sqrt{3}\right)^3=a^3+3a^2\sqrt{3}+9a+3\sqrt{3}\) \(=a^3+9a+\left(3a^2+3\right)\sqrt{3}\)

 Ta có hệ: \(\left\{{}\begin{matrix}a^3+9a=26\\3a^2+3=15\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a\left(a^2+9\right)=26\\a^2+1=5\end{matrix}\right.\)

\(\Rightarrow a=\pm2\). Nhưng rõ ràng \(a=-2\) không thỏa. Vậy \(a=2\).

 Trình bày: Ta có

 \(\sqrt[3]{26+15\sqrt{3}}\)

\(=\sqrt[3]{8+12\sqrt{3}+18+3\sqrt{3}}\)

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

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

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

Sửa đề

\(A=\left(2-\sqrt{3}\right)\sqrt[3]{26+15\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt[3]{26-15\sqrt{3}}\)

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

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

\(=\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)-\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)=0\)

Ta có: \(\left(2-\sqrt{3}\right)\sqrt{26+15\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt{26-15\sqrt{3}}\)

\(=\frac{\left(2-\sqrt{3}\right)\sqrt{52+30\sqrt{3}}-\left(2+\sqrt{3}\right)\sqrt{52-30\sqrt{3}}}{\sqrt{2}}\)

\(=\frac{\left(2-\sqrt{3}\right)\cdot\sqrt{27+2\cdot3\sqrt{3}\cdot5+25}-\left(2+\sqrt{3}\right)\sqrt{27-2\cdot3\sqrt{3}\cdot5+25}}{\sqrt{2}}\)

\(=\frac{\left(2-\sqrt{3}\right)\sqrt{\left(3\sqrt{3}+5\right)^2}-\left(2+\sqrt{3}\right)\cdot\sqrt{\left(3\sqrt{3}-5\right)^2}}{\sqrt{2}}\)

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

\(=\frac{\left(2-\sqrt{3}\right)\left(3\sqrt{3}+5\right)-\left(2+\sqrt{3}\right)\left(3\sqrt{3}-5\right)}{\sqrt{2}}\)(Vì \(3\sqrt{3}>5>0\))

\(=\frac{6\sqrt{3}+10-9-5\sqrt{3}-\left(6\sqrt{3}-10+9-5\sqrt{3}\right)}{\sqrt{2}}\)

\(=\frac{\sqrt{3}+1-\left(\sqrt{3}-1\right)}{\sqrt{2}}\)

\(=\frac{\sqrt{3}+1-\sqrt{3}+1}{\sqrt{2}}\)

\(=\frac{2}{\sqrt{2}}=\sqrt{2}\)