\(A=\sqrt{12-3\sqrt{7}}-\sqrt{12+3\sqrt{7}}\)
\(A=\dfrac{\sqrt{2}\cdot\left(\sqrt{12-3\sqrt{7}}-\sqrt{12+3\sqrt{7}}\right)}{\sqrt{2}}\)
\(A=\dfrac{\sqrt{24-6\sqrt{7}}-\sqrt{12+6\sqrt{7}}}{\sqrt{\text{2}}}\)
\(A=\dfrac{\sqrt{21-2\cdot\sqrt{21}\cdot\sqrt{3}+3}-\sqrt{21+2\cdot\sqrt{21}\cdot\sqrt{3}+3}}{\sqrt{2}}\)
\(A=\dfrac{\sqrt{\left(\sqrt{21}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{21}+\sqrt{3}\right)^2}}{\sqrt{2}}\)
\(A=\dfrac{\left|\sqrt{21}-\sqrt{3}\right|-\left|\sqrt{21}+\sqrt{3}\right|}{\sqrt{2}}\)
\(A=\dfrac{\sqrt{21}-\sqrt{3}-\sqrt{21}-\sqrt{3}}{\sqrt{\text{2}}}\)
\(A=\dfrac{-\sqrt{6}}{\sqrt{2}}\)
\(A=-\sqrt{\dfrac{6}{2}}\)
\(A=-\sqrt{3}\)
\(A=\sqrt[]{12-3\sqrt[]{7}}-\sqrt[]{12+3\sqrt[]{7}}\)
Giả sử \(\sqrt[]{12-3\sqrt[]{7}}-\sqrt[]{12+3\sqrt[]{7}}>0\)
\(\Leftrightarrow\sqrt[]{12-3\sqrt[]{7}}>\sqrt[]{12+3\sqrt[]{7}}\)
\(\Leftrightarrow12-3\sqrt[]{7}>12+3\sqrt[]{7}\)
\(\Leftrightarrow6\sqrt[]{7}< 0\left(sai\right)\)
Vậy \(\sqrt[]{12-3\sqrt[]{7}}-\sqrt[]{12+3\sqrt[]{7}}< 0\) hay \(A< 0\)
\(\Leftrightarrow A^2=12-3\sqrt[]{7}+12+3\sqrt[]{7}-2\sqrt[]{\left(12-3\sqrt[]{7}\right)\left(12+3\sqrt[]{7}\right)}\)
\(\Leftrightarrow A^2=24-2\sqrt[]{\left(144-63\right)}\)
\(\Leftrightarrow A^2=24-2\sqrt[]{81}\)
\(\Leftrightarrow A^2=24-18=6\)
\(\Leftrightarrow A=-\sqrt[]{6}\)
